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INTRODUCTION TO GEODETIC ASTRONOMY D. B. THOMSON December 1981 TECHNICAL REPORT LECTURE NOTES NO. 49217 PREFACE In order to make our extensive series of lecture notes more readily available, we have scanned the old master copies and produced electronic versions in Portable Document Format. The quality of the images varies depending on the quality of the originals. In this version, images have been converted to searchable text. INTRODUCTION TO GEODETIC ASTRONOMY Donald B. Thomson Department of Geodesy and Geomatics Engineering University of New Brunswick P.O. Box 4400 Fredericton, N.B. Canada E3B SA3 September 1978 Reprinted with Corrections and Updates December 1981 Latest Reprinting January 1997 PREFACE These notes have been written for undergraduate students in Surveying Engineering at the University of New Brunswick. The overall objective is the development of a set of practical models for the determination of astronomic azimuth, latitude and longitude that utilize observations made to celestial objects. It should be noted here that the emphasis in these notes is placed on the socalled second-order geodetic astronomy. This fact is reflected in the treatment of some of the subject matter. To facilitate the development of models, several major topics are covered, namely celestial coordinate systems and their relationships with terrestrial coordinate systems, variations in the celestial coordinates of a celestial body, time systems, timekeeping, and time dissemination. Finally, the reader should be aware of the fact that much of the information contained herein has been extracted from three primary references, namely Mueller [1969], Robbins [1976], and Krakiwsky and Wells [1971]. These, and several other's, are referenced extensively throughout these notes. D.B. Thomson. ii TABLE OF CONTENTS Page PREFACE • • • • ii LIST OF TABLES v LIST OF FIGURES vi 1. INTRODUCTION • • • • • 1.1 Basic Definitions 2. CELESTIAL 2.1 2.2 2.3 2.4 3. COORDINAT~ 1 2 SYSTEMS 14 The Celestial Sphere • • • • • Celestial Coordinate Systems • •••••• 2.2.1 Horizon System • • • • • • • • • • • • . 2.2.2 Hour Angle System ••••••••• 2.2.3 Right Ascension System. • ••• 2.2.4 Ecliptic Sys tem • • • • 2.2.5 Summary • • • • • • • • Transformations Amongst Celestial Coordinate Systems • 2.3.1 Horizon - Hour Angle • • • • • 2.3.2 Hour Angle - Right Ascension • 2.3.3 Right Ascension - Ecliptic •• 2.3.4 Summary • • • • • • • Special Star Positions 2.4.1 Rising and Setting of Stars 2.4.2 Culmination (Transit) 2.4.3 Prime Vertical Crossing. 2.4.4 Elongation • • • • • • • TIME DISSEMINATION, TIME-KEEPING, TIME RECORDING 4.1 Time Dissemination • • • • • • • 4.2 Time-Keeping and Recording. 4.3 Time Observations • • • • • 5. 26 29 32 34 36 42 44 47 47 50 59 61 64 TIME SYSTEMS 3.1 Sidereal Time • • • • • • 3.2 Universal (Solar) Time • • • 3.2.1 Standard(Zone) Time 3.3 Relationships Between Sidereal and Solar Time Epochs and Intervals •• • • • • • 3.4 Irregularities of Rotational Time Systems • 3.5 Atomic Time System. • • • • 4. 14 20 20 23 STAR CATALOGUES AND EPHEMERIDES 5.1 Star Catalogues .5.2 Ephemerides and Almanacs •• iii 67 75 80 80 93 93 .... 95 . . •. 95 99 • • • 100 ... • • • • 106 • 108 Page 6. 7. 8. 9. VARIATIONS IN CELESTIAL COORDINATES· • • • • • • • • • • 117 6.1 Precession, Nutation, and Proper Motion· 6.2 Annual Aberration and Parallax • • • • • • • • • • 6.3 Diurnal Aberration, Geocentric Parallax, and • • •• •••• Astronomic f'1.~(~~'~:.ton 6.4 Polar Motion • • • • • • • • • • • • • 118 127 DETERMINATION OF ASTRONOMIC AZIMUTH 134 138 ........ 147 7.1 Azimuth by Star Hour Angles •• 7.2 Azimuth by Star Altitudes 148 154 DETERMINATION OF ASTRONOMIC LATITUDE • 163 8.1 Latitude by Meridian Zenith Distances •• 8.2 Latitude by Polaris at any Hour Angle •• 164 165 DETERMINATION OF ASTRONOMIC LONGITUDE • • • 170 9.1 Longitude by Meridian Transit Times· • REFERENCES • • • • • • • • • • • • • • • • • • 172 • 175 APPENDIX A Review of Spherical Trigonometry •• 176 APPENDIX B Canadian Time Zones 183 APPENDIX C Excerpt from the Fourth Fundamental Catalogue (FK4) APPENDIX D . . • . . • . . • . • .,. . • ... . . 184 Excerpt from Apparent Places of Fundamental Stars •• • • • • • • • • • • • • • • • • • • • • • • 186 iv LIST OF TABLES Table 2-1 Celestial Coordinate Systems [Mueller, 1969] . • 33 Table 2-2 Cartesian Celestial Coordinate Systems [Mueller, 1969] 33 Table 2-3 Transformations Among Celestial Coordinate Systems [Krakiwsky and Wells, 1978J . Table 6-1 Description of Terms • . Table 7-1 Azimuth by Hour Angle:Random Errors in Latitude 60 0 133 155 [Robbins, 1976] Table 7-2 49 Azimuth by Altitude: Random Errors in Latitude 60 0 [Robbins, 1976] • • • • • • v 158 LIST OF FIGURES Figure 1-1 Biaxial Ellipsoid • . • • • • 4 Figure 1-2 Geodetic Latitude, Longitude, and Ellipsoidal Height 5 Figure 1-3 Orthometric Height 6 Figure 1-4 Astronomic Latitude Figure 1-5 Geoid Height & Terrain Deflection of the Vertical • . . Figure 1-6 Components of the Deflection of the Vertical 10 Figure 1-7 Geodetic Azimuth • • 12 Figure 1-8 Astronomic Azimuth 13 Figure 2-1 Cele,stial Sphere 16 Figure 2-2 Astrono~ic Figure 2-3 Sun's Apparent Motion Figure 2-4 Horizon Figure 2-5 Horizon System Figure 2-6 Hour Angle System. 24 Figure 2-7 Hour Angle System 25 Figure 2-8 Right Ascension System 27 Figure 2-9 Right Ascension System 28 (~) and Longitude (A) 8 •• Triangle. . 9 17 ....... System . . . . . . . . . . .... .... .... 19 21 22 Figure 2-10 Ecliptic System. 30 Figure 2-11 Ecliptic System. 31 Figure 2-12 Astronomic Triangle • 35 Figure 2-13 Local Sidereal Time • 37 vi List of Figures Cont'd Figure 2-14 Horizon and Hour Angle System Transformations • 39 Figure 2-15 Horizon 41 Figure 2-16 Hour Angle - Right Ascension Systems • 43 Figure 2-17 Right Ascension-Ecliptic Systems • • 45 Figure 2-18 Celestial Coordinate System Relationships [Krakiwsky and Hour Angle Systems • • • • • • • and We.lls, 1971] •. 48 Figure 2-19 Circumpolar and Equatorial Stars • 51 Figure 2-20 Declination for Visibility • • 52 Figure 2-21 Rising and Setting of a Star • • 54 Figure 2-22 Hour Angles of a Star's Rise and Set 55 Figure 2-23 Rising and Setting of a Star Figure 2-24 Azimuths of a Star's Rise and Set. 58 Figure 2-25 Culmination (Transit) • • 60 Figure 2-26 Prime Vertical Crossing • 62 Figure 2-27 Elongation. • 65 Figure 3-1 Sidereal Time Epochs • 70 Figure 3-2 Equation of Equinoxes (OhUT , 1966) [Mueller, 1969] 71 Figure 3-3 Sidereal Time and Longitude 72 Figure 3-4 Universal and Sideral Times fAA, 1981] • 74 Figure 3-5 Universal (Solar) Time • • • . • • • • • 76 Figure 3,...6 Equation of Time (Oh UT , 1966) [Mueller, 1969] • 78 Figure 3-7 Solar Time and Longitude • • • 79 Figure 3-8 Standard (Zone) Time [Mueller, 1969] 81 Figure 3-9 Relationships Between Sidereal and Universal Time Epochs 83 vii .. .. ... .. 57 List of Figures Cont'd Figure 3-10 [SALS~-.i98ir. Sun - February. 1978 • • 86 Figure 3-11 Mutual Conversion of Intervals of Solar and .Siderea1 Time [SALS, 19811 • 87 Figure 3-12a Universal and Sidereal Times, ·1981 [AA,1981J • 88 Figure 3-12b Universal and Sidereal Times, 1981 cont'd. [AA, 1981] • • • • • • • • • 89 Figure 3-12c Universal and Sidereal Times, 1981 cont'd. 90 Figure 4-1 Code for the Transmission of DUT1 97 ~econd... Figure 4-2 ,:·pating. of Events in the Vicinity of A Leap Figure 5-1 Universal and Sidereal Times, 1981 [AA ,1981' . J Figure 5-2 Bright Stars, 1981.0 [AA,··r9S1] ......... ···· .··. .,- . 98 109 110 .. Figure 5-3 Bright Stars, 1981.0 cont'd. [AA" 1281] Figure 5-4 Sun - February, 1981 [SALS ,1981 L .. Figure 5-5a Right Ascension of Stars, 1981 [BALS, 1981 ] Figure 5-5b Declination of Stars t 1981, [SALS, 1981].· Figure 5-6 Pole Star Table, 1981 [SALS, 1-981] Figure 6-1 Variations of the Right Ascension System • Figure 6-2 Motion of the Celestial Pole 121 Figure 6-3 Mean Celestial Coordinate 123 Figure 6-4 True and Mean Celestial coordinate Systems Figure 6-5 Aberration Figure 6.;;.;6 Annual (Stellar) Parallax •• Figure 6-7 Position Updating M.Co T to A.Po T ./ .. 0 . 0 0 0 Geocentric Parallax 0 .... viii . . .. ·· . 0 0 0 · 115 .' ···· 0 0 0 . · 0 0 0 0 0 116 119 126 ···. 0 113 114 . . . · · · . . . .. Systems . ·..·.···· 0 Figure 6-8 III 129 130 0 0 ..·..· 0 132 0 0 0 0 0 0 136 List of Figures cont'd Figure 6-9 Astronomic Refraction • • • • • • • • • • . 137 Figure 6-10 Observed to Apparent Place 139 Figure 6-11 Transformation of Apparent Place 141 Figure 6-12 Transformation from Instantaneous to Average Terrestrial System • • • • Figure 6-13 143 Celestial Coordinate Systems and Position Updating [Krakiwsky, Wells, 1971] • . ••• 146 Figure 6-:-14 Coordinate Systems Used in Geodesy • Figure 7-1 Azimuth by the Hour Angle of Polaris [Mueller, 1969] 152 Figure 7-1 Cont'd • • 153 Figure 7-2 Azimuth by the Sun's Altitude • • • • • • • • • • • . • 159 ix •••• 145 1. INTRODUCTION Astronomy is defined as study of the universe beyond the [Morris~ earth~ 19751 "The scientific especially the observation, calculation, and theoretical interpretation of the positions, dimensions, distribution~ phenomena". motion, composition, and evolution of celestial bodies and Astronomy is the oldest of the natural sciences dating back to ancient Chinese and Babylonian civilizations. Prior to 1609, when the telescope was invented, the naked eye was used for measurements. Geodetic astronomy, on the other hand, is described as [Mueller, 1969] the art and science for determining, by astronomical observations, the positions of points on the earth and the azimuths of the geodetic lines connecting such points. When referring to its use in surveying, the terms practical or positional astronomy are often used. The fundamental concepts and basic principles of "spherical astronomy", which is the basis for geodetic astronomy, were developed principally by the Greeks, and were well established by the 2nd century A.D. The treatment of geodetic astronomy in these notes is aimed at the needs of undergraduate surveying engineers. To emphasise the needs, listed below are ten reasons for studying this subject matter: (i) a knowledge of celestial coordinate systems, transformations amongst them, and variations in each of them; (ii) celestial coordinate systems define the "linkll between satellite and terrestrial coordinate systems; (iii) the concepts of time for geodetic purposes are developed; (iv) tidal studies require a knowledge of geodetic astronomy; 1 2 (v) when dealing with new technologies (e.g. inertial survey systems) an understanding of the local astronomic coordinate system is essential; (vi) astronomic coordinates of terrain points, which are expressed in a "natural" coordinate system, are important when studying 3-D terrestrial networks; (vii) astrondmically determined azimuths provide orientation for terrestrial networks; (viii) the determination of astrogeodetic deflections of the vertical are useful for geoid determination, which in turn may be required for the rigorous treatment of terrestrial observations such as distances, directions, and angles; (ix) geodetic astronomy is useful for the determination of the origin and orientation of independent surveys in remote regions; (x) geodetic astronomy is essential for.the demarcation of astronomically defined boundaries. 1.1 Basic definitions In our daily work as surveyors, we commonly deal with three different surfaces when referring to the figure of the earth: (i) the terrain, (ii) an ellipsoid, and (iii) the geoid. The physical surface of the earth is one that is extremely difficult to model analytically. It is common practice to do survey computations on a less complex and modelable surface. The terrain is, of course, that surface on or from which all terrestrially based observations are made. The most common figure of the earth in use, since it best approximates the earth's size and shape is a biaxial ellipsoid. It is a purely 3 mathematical figure, defined by the parameters a (semi-major axis) b (semiminor axis) or a and f (flattening), where f=(a-b)/a (Figure 1-1). This figure is commonly referred to as a "reference ellipsoid", but one should note that there are many of these for the whole earth or parts thereof. The use of a biaxial ellipsoid gives rise to the use of curvilinear geodetic coordinates <P (latitude), A(longitude) ,and h (ellipsoidal height) (Figure 1-2). Obviously, since this ellipsoid is a "mathematical" figure of the earth, its position and orientation within the earth body is chosen at will. Conventionally, it has been positioned non-geocentrically, but the trend is now to have a geocentric datum (reference ellipsoid). Conventional orientation is to have parallel~m of the tertiary (:Z) axis with the mean rotation axis of the earth, and parallelism' of the primary (X) axis with the plane of the Greenwich Mean Astronomic Mer:t.dian. Equipotential surfaces (Figure 1-3), of which there are an infinite number for the earth, can be represented mathematically. physical properties They account for the of the earth such as mass, mass distribution, and rotation, and are "real" or physical surfaces of the earth. The common equipotential surface used is the geoid, defined as [Mueller, 1969] " t hat equipotential surface that most nearly coincides with the undisturbed mean surface of the oceans". (Figure Associated with these equipotential surfaces is the plumhline 1~3). It is a line of force that is everywhere normal to the equipo- tential surfaces; thus, it is a spatial curve. We now turn to definitions of some fundamental quantities in geodetic astronomy namely, the astronomic latitude (4)), astronomic longitude (A), and orthometricheight (H). These quantities are sometimes referred to as "natural" coordinates, since, by definition, they are given in terms of the ·4 z p pi Figure 1-1 Biaxial Ellipsoid 5 .. z. ELLIPSOIDAL NORMAL ~~-r------~~Y o· (A=90) Figure 1-2 Geodetic Latitude,Longitude, and r.llipsoida1Height , 6 PLUMBLINE EQUIPOTENTIAL SURFACES TERRAIN GEOID r ELL~ Figure 1-3 Orthometric lIeieht 7 "real" (physical) properties of the earth. Astronomic latitude (~) is defined as the angle between the astronomic normal (gravity vertical) (tangent to the plumb1ine a.t the point of interest) and the plane of the instantaneous equator measured in the astronomic meridian plane (Figure 1... 4). Astronomic longitude (/I.) is the angle between the Greenwich Mean Astronomic Meridian and the astronomic meridian plane measured in the plane of the instantaneous equator (Figure 1... 4). The orthometr1c height (H), is the height of the point of interest .above the geoid, measured along the plumbline, as obtained from spirit leveling and en route gravity observations (Figure 1-3). reductions of ~ Finally, after some and A for polar motion and plumb line curvature, one obtains . the "reduced" astronomic coordinates (~, A, H) referring to the geoid and the mean rotation axis of the earth (more will be said about this last point in these notes). We are now in a position to examine the relationship between the Geodetic and Astronomic coordinates. This is an important step for surveyors. Observations are made in the natural system; astronomic coordinates are expressed in the same natural system; therefore, to use this information for computations in a geodetic system, the relationships must be known. The astro-geodetic (relative) deflection of the vertical (a) at a point is the angle between the astronomic normal at that point and the normal to the reference ellipsoid at the corresponding point (the point may be on the terrain (at) t~o components, or on the geoid{s g ) (Figure 1-5). ~ n- - meridian and e is normally split into prime vertical (Figure 1-6). Mathema- tically, the components are given by ~ n = ~- !fl, = (A-A) cos !fl, (1-1) (1-2) 8 TRUE. - Z'T' ROTATION 'AXIS INSTANTANEOUS POLE PlUMBliNE GRAVITY VERTICAL Figu.re 1-4 Astronomic Latitu.de (4)) and Longitude (A) 9 ELLIPSOIDAL PLUMBLINE NORMAL-~ EQUIPOTENTIAL SURFACE' ~~~-- - -- "'ERRA.IN GEOID ELLIPSOID Fieure 1-5 Geoid Ueieht & Terrain Deflection of the Vertical z liZ 10 liZ 0 ELLIPSOIDAL NORMAL UNIT SPHERE 7J =(1\. - 'A.) cos <P Figure 1-6 Components of the Deflection of the Vertical 11 which yields the geodetic-astronomic coordinate relationships we were seeking. The geoida1 height (N) is the distance between the geoid and a reference ellipsoid, measured along an ellipsoidal normal (Figure 1-5). Mathematically, N is given by (with an error of less than 1mm) tie1sianen and Moritz, 1967] N = h-H. (1-3) Finally, we turn our attention to the azimuths of geodetic lines between points. A geodetic azimuth (a), on the surface of a reference ellipsoid, is the clockwise angle from north between the geodetic meridian of i and the tangent to the ellipsoidal surface curve of shortest distance (the geodesic) between i and j (Figure 1-7). The astronomic azimuth (A) is the angle between the astronomic meridian plane of i and the astronomic normal plane of i through j (Figure 1-8), measured clockwise from north. The relationship between these A and a Laplace Azimuth equation [e.g. Heiskanen and (A-a) = ntan~ + (~sina in which z is the zenith distance. Moritz~ - ncosa) cot z, is given by the 1967] (1-4) Note that the geodetic azimuth, a, must also be corrected for the height of target (skew-normal) and normal section geodesic separation. 12 ZG #4-----'--4- GEODES I C kc::--t----=~-----l~-l-- XG Figure 1-7 Geodetic Azimuth YG 13 ZLA Z AT X CIO LA ~ASTRONOMIC . NORMAL NORMAL PLANE OF THROUGH j VLA , ASTRONOMIC' MERIDIAN Figure 1-R Astronomic Azimuth 2. CELESTIAL COORDINATE SYSTEMS Celestial coordinate systems are used to define the coordinates of celestial bodies such as stars. There are four main celestial coordinate systems - the ecliptic,'rightascelision, hour angle, and horizon of which are examined in detail in this chapter. each There are two fundamental differences between celestial coordinate systems and terrestrial and orbital coordinate systems. First, as'a consequence of the great distances involved, only directions are considered in celestial coordinate sytems. This means that all vector quantities dealt with can be considered to be unit vectors. The second difference is that celestial geometry is spherical rather than ellipsoidal which simplifies the mathematical relationships involved. 2.1 The Celestial Sphere The distance from the earth to the nearest star is more than 10 9 earth radii, thus the dimensions of the earth can be considered as negligible compared to the distances to the stars. For example, the closest star is estimated to be 4 light years (40xl0 12 km) from the earth (a CENTAURI), while " others are VEGA at 30 light years, a USAE MINORIS (Polaris) at 50 light years. Our sun is only 8.25 light minutes (155xl0 6 km) from the earth. As a consequence of those great distances, stars, considered to be moving at near the velocity of light, are perceived by an observer on earth to be moving very little. Therefore, the relationship between the earth and stars can be closely approximated by considering the stars all to be equidistant from the earth and lying on the surface of a celestial sphere, the dimension of which is so large that the earth, and indeed the solar system, can be considered to be a dimensionless point at its centre. 14 Although this point may be 15 considered dimensionless, relationships between directions on the earth and in the solar system can be extended to the· celestial sphere. The instantaneous rotation axis of the earth intersects the celestial sphere at the north and south celestial poles (NCP and SCP respectively) (Figure 2-1). The earth's equitorial plane extented outwara intersects the celestial sphere at the celestial equator (Figure 2-1). The vertical (local-astronomic normal) intersects the celestial sphere at a point above the observer, the zenith, and a point beneath. the observer, the nadir (Figure 2-1). A great circle containing the poles, and is thus perpendicular to the celestial equator, is called an hour circle (Figure 2-1). The observer's vertical plane, containing the poles, is the hour circle through the zenith and is the observer's 'celestial meridian (~igure 2-1). A small circle parallel to the celestial equator is called a celestial parallel. Another very important plane is that which is normal to the local astronomic vertical and contains the observer (centre of the celestial sphere); celestial horizon (Figure 2-1). The plane normal to the horizon passing through the zenith is the vertical plane. celestial horizon is called an it is the almucantar~ A small circle parallel to the The vertical plane normal to the celestial meridian is called the prime vertical. The intersection points of the prime vertical and the celestial horizon are the ~ and west points. Due to the rotation of the earth, the zenith (nadir), vertical planes, almucantars, the celestial horizon and meridian continuously change their positions on the celestial sphere, the effects of which will be studied later. If at any instant we select a point S on the celestial sphere (a star), then the celestial meridian and the hour and vertical circles form a spherical triangle called the astronomic triangle of S. Its verticies are the zenith 16 NCP I .I . . I . SCP Figure 2-1 Celestial S .. p h ere 17 I - I - . Figure 2-2 A~tronoIlU.. c Triangle 18 (Z), the north celestial pole (NCP) and S(Figure 2-2). There are also some important features on the celestial sphere related the revolution of the earth about the sun, or in the reversed concept, the apparent motion of the sun about the earth. The most important of these is the ecliptic, which may be described as the approximate (within 2" [Mueller, 1969]) apparent path of the sun about the earth.(Figure 2-3). The ecliptic intersects the celestial equator in a line connecting theeqtiinoxes. The vernal equinox is that point intersection where the apparent sun crosses the celestial equator from south to north. equinox. The other equinox is the autumnal The acute angle between the celestial equator and the ecliptic is termed the obliquity of the ecliptic (e:=23°27') (Figure 2-3). The points at 90 0 from either equinox are the points where the sun reaches its greatest angular distance from the celestial equator, and they are the summer solstice (north) and winter solstice (south). In closing this section we should note that the celestial sphere is only an approximation of the true relationship between the stars and an observer onthe earth's surface. Like all approximations, a number of corrections are required to give a precise representation of the true relationship. corrections represent the facts that: These the stars are not stationary points on the celestial sphere but are really moving (proper motion); the earth's rotation axis is not stationary with respect to the stars (precesion, nutation); the earth is displaced from the centre of the celestial sphere (which is taken as the centre of the sun) and the observer is displaced from the mass centre of the earth (parallax); the earth is in motion around the centre of the celestial sphere (aberration); and directions measured through the earth's atmosphere are bent by refraction. All of these effects are discussed in 19 I AUTUMNAL I .". .". _ ~QLEN~X\_I __ .... .". '" SUMMER SOLSTICE - --- "'" I .~ 1\ \-----o::~--I-OBL IQ U ITV ---~~~~V~E-R-NAL I EQU INOX (¥) t I I I SCP . Figure 2-3 Sun's Apparent Motion OF THE ECLIPTIC 20 detail in these notes. 2.2 Celestial Coordinate Systems Celestial coordinate systems are used to define the positions of stars on the celestial sphere. Remembering that the distances to the stars are very great, and in fact can be considered equal thus allowing us to treat the celestial sphere as a unit sphere, positions are defined by directions only. One component or curva1inear coordinate is reckoned from a primary reference plane and is measured perpendicular to it, the other from a secondary reference p1an~· and is measured in the primary plane. In these notes, two methods of describing positions are given. The first is by a set of curvalinear coordinates, the second by a unit vector in three dimensional space expressed as a function of the curva1inear coordinates. 2.2.1 Horizon System The primary reference plane is the celestial horizon, the secondary is the observer's celestial meridian (Figure 2-4). This system is used to describe the position of a celestial body in a system peculiar to a topographically located observer. The direction to the celestial body S is defined by the a1titude(a) and azimuth (A) (Figure 2-4). The altitude is the angle between the celestial horizon and the point S measured in the plane of the vertical circle (0° - 90°). zenith distance. The complimentary angle z =90-a, is called the The azimuth A is the angle between the observer's celestial meridian and the vertical circle through S measured in a clockwise direction (north to east) in the plane of the celestial horizon (0° - 360°). 21 ~~NORTH POINT Figure 2-4 Ilorizon System 22 NCP ZH I I I Z S X x ~SIN a=Z/1 :o-..~SIN A=Y/COS A_ ~ Y a ." A=X/COS a Figure 2-5 J!orizon System ZH COS a . COS a COS A SIN A SIN a 23 To determine the unit vector of the point S in terms of a and A, we must first define the origin and the three axes of the coordinate system. The origin is the he1iocentre (centre of mass of the sun [e.g. Eichorn, 1974]~. The primary pole (Z) is the observer's zenith (astronomic normal or gravity vertical). The primary axis (X) is directed towards the north point. The secondary (Y) axis is chosen so that the system is left-handed (Figure 2-5 illustrates this coordinate system). Note that although the horizon system is used to describe the position of a celestial body in a system peculiar to a topographically located observer, the system is heliocentric and not topocentric. The unit vector describing the position of S is given by cosa [ cosa COSAJ sinA (2-1) sina Conversely, a and A, in terms of [X,Y,Z] a = sin-1 Z, A = tan 2.2.2 -1 Y T are (Figure 2-5) (2-2) (IX). (2-3) Hour Angle System The primary refer~nce plane is the celestial eguator, the secondary is the hour circle containing the zenith (obserser's celestial meridian). The direction to a celestial body S on the celestial sphere is given by the declination (0) and hour angle (h). The declination is the angle between the ce1ectia1 equator and the body S, measured from 0 0 to 90 0 in the plane of the hour circle through S. polar distance. * The complement of the declination is called the The hour angle is the angle between the hour circle of S Throughout these notes, wemake'the valid approximation heliocentre barycentre of our solar system. = 24 NCP Figure 2-6 Hour Angle System 25 ZHA XHA x Y Z = HA COS 6 COS h COS 6 SIN h SIN 6 Figure 2-7 Hour Angle System 26 and the observer's celestial meridian (hour circle), and is measured from oh h to 24 , in a clockwise direction (west, in the direction of the star's apparent daily motion) in the plane of the celestial equator. Figure 2-6 illustrates the hour angle system's curva1inear coordinates. To define the unit vector of S in the hour angle system, we define the coordinate system as follows (Figure 2-6). The origin is the he1iocentre. The primary plane is the equatorial plane, the secondary plane is the celestial meridian plane of the observer. The primary pole (Z) is the NCP, the primary axis (X-axis) is the intersection of the equatorial and observer's celestial meridian planes. The Y-axis is chosen so that the system is left-handed. The hour angle system rotates with the observer. XL [Y • Z and ~ [coso: COSh] cos& sinh sinS , (2-4) and h are given by o = sin- 1 Z -1 h = tan 2.2.3 The unit vector is , (2-S) (Y/x). (2-6) Right Ascension System The right ascension system is the most important celestial system as it is in this system that star coordinates are published. It also serves as the connection between terrestrial, celestial, and orbital coordinate systems. The primary reference plane is the celestial equator and the secondary is the equinocta1 co1ure (the hour circle passing through the NCP and SCP and the vernal and autumnal equinoxes){Yigure 2-8). The direction to a star S is given by the right ascension (a) and declination (6), the latter of which 27 NCP Figure 2-8 Right Ascension System 28 ZRA YRA XRA COS 6 COS COS 6 SIN SIN 6 Figure 2-9 Ri~ht Ascension System ~ OC 29 has already been defined (RA system). The right ascension is the angle between the hour circle of S and the equinoctal colure, measured from the vernal equinox to the east (counter clockwise) in the plane of the celestial h equator from 0 h to 24 • The right ascension system coordinate axes are defined as having a heliocentric origin, with the equatorial plane as the primary plane, the primary pole (Z) is the NCP, the primary axis (X) is the vernal equinox, and the Y-axis is chosen to make the system right-handed (Figure 2-9). The unit vector describing the direction of a body in the right-ascension system is : ... cos' 0 cos<~ X - Y Z RA cos 15 sino. , (2-7) sino and a and 0 are expressed as -1/<" (Y!X) , a - tan 15 = sin-lz (2-8) (2-9) 2.2;4 Ecliptic System The ecliptic system is the celestial coordinate system that is closest to being inertial, that is, motionless with respect to the stars. However, due to the effect of the planets on the earth-sun system, the ecliptic plane is slowly rotating (at '0".5 per year) about a slowly moving axis of rotation. The primary reference plane is the ecliptic, the secondary reference plane is the ecliptic meridian of the vernal equinox (contains the north and south ecliptic poles, the vernal and autumnal equinoxes) (Figure 2-10). The direction 30 SCP Figure 2-10 Ecliptic System 31 x COS ~ COS A.. COS ~ SINA. Y Z E SIN ~ Figure 2-11 :. Ecliptic System 32 to a point S on the celestial sphere .is given by the ecliptic latitude (B) and ecliptic longitude (A). The ecliptic latitude is the angle, measured in the ecliptic meridian plane of S, between the ecliptic and the normal OS (Figure 2~10). The ecliptic longitude is measured eastward in the ecliptic plane between the ecliptic meridian of the vernal equinox and the ecliptic meridian of S (Figure 2-10). The ecliptic system coordinate axes are specified as follows (Figure 2-11). The origin is heliocentric. The primary plane is the ecliptic plane and the primary pole (Z) is the NEP (north ecliptic pole). The primary axis (X) is the normal equinox, and the Y-axis is chosen to make the system right-handed. The unit vector to S is x = Y cosB COSA cosS sinA Z , (2-10) sinS while B and A are given by 2.2.5 -1 S = sin A = tan -1 Z, (2-11) (y/X) (2-12) Summary The most important characteristics of the coordinate systems, expressed in terms of curva1inear coordinates, are given in Table 2-1. The most important characteristics of the cartesian coordinate systems are shown in Table 2-2 (Note: ~ and u in Table 2-2 denote the curvilinear coordinates measured in the primary reference plane and perpendicular to it respectively). 33 Parameters Measured from the Reference Plane System Primary Secondary Primary Secondary Horizon Celestial horizon Celestial meridian (half Alti.tude -90~!:a !:+900 containi.ng north pole) (+toward zenith) Hour Angle Celestial equator Hour circle of observer's zentth (half containing zenith) Azimuth 00<A<3600 (+east) Hour Declination -90°<15< +90° (+north) an~le Oh~h~24h 00~h~360° (-+west) Right Ascension Celestial equator Equinoctical colure (half containing vernal equinox) Declination -90°s.15s.+90° (+north) \ Right Ascen sion Oh<a.<24h 00~a.~360° (+east) Ecliptic Ecliptic Ecliptic meridan equinox (half containing vernal equinox) (Ecliptic) Longitude 0°<:>'<360° (Ecliptic) Latitude 90°<6<+90° (+~a~t) (+n~rth) CELESTIAL COORDINATE SYSTEMS [Mueller, 1969] TABLE 2-1 System Horizon Hour angle Orientation of the Positive Axis X Y Z (Secondary pole) (Primary pole) u A a left North point A .,. 90° Intersection of the zenith's hour circle with the celestial equator on the zenith's side. h = 90°. 6h North celestial h pole 15 left a. .,. 90°= 6h North celestial a. pole 15 right 6 right Right ascension Vernal equinox Ecliptic Left or Right handed )J Vernal equinox = 90° Zenith North ecliptic pole :>. CARTESIAN CELESTIAL COORDINATE SYSTEMS [Mueller, 1969] TABLE 2-2 34 2.3 Transformations Amongst Celestial Coordinate Systems Transformations amongst celestial coordinate systems is an important aspect of geodetic astronomy for it is through the transformation models that we arrive at the math models for astronomic position and azimuth determination, Two approaches to coordinate systems transformations are dealt with here: (i) the traditional approach, using spherical trigonometry, and (ii) a'more general approach using matrices that is particularly applicable to machine computations. The relationships that are developped here are between (i) the horizon and hour angle systems, (ii) the hour angle and right ascension systems, and (iii) the right ascension and ecliptic systems. With these math models, anyone system can then be related to any other system (e.g. right ascension and horizon systemS). Before developing the transformation models, several more quantities must be defined. To begin with, the already known quantities relating to the astronomic triangle are (Figure 2-12): (i) from the horizon system, the astronomic azimuth A and the altitude a or its compliment the zenith distance z=90-a, (ii) from the hour angle system, the hour angle h or its compliment 24-h, (iii) from the hour angle or right ascension systems, the declination 0, or its compliment, the polar distance 90-0. The new quantities required to complete the astronomic triangle are the astronomic latitude longitude ~ or its compliment 90-~, the difference in astronomic 6A =As- Az (=24-h), and the parallactic angle p defined as the angle between the vertical circle and hour circle at S (Figure 2-12). The last quantity needed is Local Sidereal Time. (Note: this is 35 EQUATOR Figure 2-12 Astronomic Triangle 36 not a complete definition, but is introduced at this juncture to facilitate coordinate transformations. A complete discussion of sidereal time is presented in Chapter 4). To obtain a definition of Local Sidereal Time, we look at three meridians: (i) the observer's celestial meridian, (ii) the hour circle through S, and (iii) the equinoctal colure. from the NCP, we see, 2-13): Viewing the celestial sphere on the equatorial plane, the following angles (Figure (i) the hour angle (h) between the celestial meridian and the hour circle (measured clockwise), (ii) the right ascension (a) between the equinoctal colure and the hour circle (measured counter clockwise), and (iii) a new quantity, the Local Sidereal Time (LST) measured clockwise from the celestial meridian to the equinoctal colure. LST is defined as the hour angle of the vernal equinox. 2.3.1 Horizon - Hour Angle Looking first at the Hour Angle to Horizon system transformation, using a spherical trigonometric approach, we know that from the hour angle system we are given h and ~, and we must express these quantities as functions of the horizon system directions, a (or z) and A. formation is a knowledge of Implicit in this trans- ~. From the spherical triangle (Figure 2-14), the law of sines yields sin(24-h) sinz = sinA ~~77~~ sin(90-~) , (2-13) or -sinh sinz sinA = ---cos~ (2-14) and finally sinA sinz = -sinh cos~ (2-15) 37 ....JZ <! ~9 wr:t: ....J w W:E u Figure 2-13 Local Sidereal Time 38 The five parts formula of spherical trigonometry gives cosA sinz = cos(90-~)sin(90-~)-cos(90-~)sin(90-~)cos(24-h), (2-16) or cosA sinz = sin6 - cos~ sin~ cos6 cosh (2-17) Now, dividing (2-15) by (2-17) yields sini\. sinz cosA sinz -sinh cos~ sin6 cos~ -sin~ coso cosh = (2-18) which, after cancelling and collecting terms and dividing the numerator and denominator of the right-hand-side by coso yields tanA = -sinh tan~ cos~ -sin~ , cosh (2-19) or tanA = sin~ sinh cosh - tano cos~ (2-20) Finally, the cosine law gives cosz = cos(90-~)cos(90-9) + sin(90-~)sin(90-9)COs(24-h), (2-21) + (2-22) or cost = sin~ sino cos~ coso cosh Thus, through equations (2-20) and (2-22), we have the desired results - the quantities a (z) and A expressed as functions of 0 and h and a known latitude ~. The transformation Horizon to Hour Angle system (given a (=90-z), A, ~, compute h, 0) is done in a similar way using spherical trigonometry. The sine law yields coso sinh = -sinz sinA , (2-23) sin~ (2-24) and five parts gives, coso cosh = cosz cos~ -sinz cosA 39 EQUATOR Figure 2-14 llorizonand Hour Angle System Transformations 40 After dividing (2-23) by (2-24), the result is __~__s7i=nA~________~ tanh _ cosA -cotz cos~ + sinz cosA cos~ sin~ (2-25) Finally,the cosine law yields sineS = cosz sin~ (2-26) Equations (2-25) and (2-26) are the desired results - hand eS expressed solely as functions of a(=90-z),A, and ~. Another approach to the solution of these transformations is to use rotation matrices. handed. First, both systems are heliocentric and left- From Figure 2-15, we can see that YHA is coincident with -YH (or YH coincident with -YHA), thus they are different by 180°. 2-15, ZH ~, an~ ~ ZH' ZHA' and ~ Also from Figure all lie in the plane of the celestial meridian. are also separated by (90-~) and are different by 180°. The transformation, then, of HA to H is given simply by x x y = (2-27) Z H Z HA or giving the fully expanded form of each of the rotation matrices X Y Z = cos180° sin1800 0 -s.in1800 cos180° 0 0 H 1 0 cos(90-~) 0 sin(90-~) 0 -sin(90-~) X 1 0 Y 0 cos (90-11» . (2-28) ZHA The effect of the first rotation, R2 (90-11», is to bring ZHA into coincidence with ZH' and ~ into the same plane as rotation, R3 (1800), brings ~ ~ (the horizon plane). and YHA into coincidence with respectively, thus completing the transformation. ~ The second and YH 41 NCP EAST POINT Figure 2-15 Angle Systems Horizon and Hour . 42 The reverse transformation - Horizon to Hour Angle - is simply the inverse* of (2-27), namely = (2-29) H 2.3.2 Hour Angle - Right Ascension Dealing first with the spherical trigonometric approach, it is evident from Figure 2-16 that LST = h + a. (2-30) Furthermore, in both the Hour Angle and Right Ascension systems, the declination 0 is one of the star coordinates. To transform the Hour Angle coordinates to coordinates in the Right Ascension system (assuming LST is known) we have a = LST-h (2-31) (2-32) 0=0 Similarly, the hour angle system coordinates expressed as functions of right aS,cension system coordinates are simply h = (2-33) LST-a 0=0 (2-34) For a matrix approach, we again examine Figure 2-16. are heliocentric, ZHA= ZRA' and *Note: ~, YHA ,' ~, Note that both systems and YRA all lie in the plane For rotation matrices, which are orthogonal, the .. f~t1owing rules apply. If x=Ry, then y=R-1x ; also (Ri Rj )-l = ~lRi' and 'R- 1 '(S) == R(-S). 43 Figure 2-16 Hour Ang1e·- Right Ascension Systems 44 of the celestial equator. The differences are that the HA system is left- handed, the RA system right-handed, and LST. XHA and ~ are separated by the The Right Ascension system, in terms of the hour angle system, is given by (2-35) or, with R3 (-LST) and the ,reflection matrix P2 expanded, [x] [ ~ ~] [~ cos(-LST) sin(-LST) ~ -Sin(~LST) COS(~LST) o (2-36) -1 o The reflection matrix, P2 , changes the handedness of the hour angle system (from left to right), and the rotation R3 (-LST) brings cidence with ~ XHA and YHA into coin- and YRA respectively. The inverse trans~ormation is simply (2-37) 2.3.3 Right Ascension - Ecliptic For the spherical trigonometric approach to the Right Ascension - Ecliptic system transformations, we look at the spherical triangle with vertices NEP, NCP, and S in .Figure B, 6 2~17. The transformation of the Ecliptic coordinates (€ assumed known) to the Right Ascension coordinates a, 0, utilizes the same procedure as in 2.3.1. 45 Z RA SUMMER SOLSTICE YE -o h:. .I ~ CELESTIAL Figure 2-17 ystems Right Ascension-Ec1ipti c S' · 46 The sine rule of spherical trigonometry yields coso cosa == cos~ (2-38) cosA and the five parts rule coso sina = cos~ (2-39) sinA COSE - sinS sinE Dividing (2-39) by (2-38) yields the desired result tana sinA COSE cosA = tanS SinE (2-40) From the cosine rule sino = cos~ sinA sinE + sinS COSE , (2-41) which completes the transformation of the Ecliptic system to the Right Ascension system. Using the same rules of spherical trigonometry (sine, five-parts, cosine), the inverse transformation (Right Ascension to Ecliptic) is given by cosS co.sA == coso cosa cosS sinA = coso , sina COSE + sino sinE , (2-42) (2-43) which, after dividing (2-43) by (2-42) yields tanA == sina COSE + tano sinE cosa (2-44) and sinS = -coso sina sinE + sino COSE • (2-45) From Figure 2-17, it is evident that the difference between the E and RA cartesian systems is simply the obliquity of the ecliptic, E, which separates the ZE and ZRA' and YE and YRA axes, the pairs of which lie in the same planes. 47 The transformations then are given by [:lll = R1 (-e:) [~L [:] == Rl(e:) [~] (2-46) and 2.3.4 E (2-47) II Summary Figure 2-18 and Table 2-3 summarize the transformations amongst celestial coordinate systems. Note that in Figure 2-18, the quantities that we must know to effect the various transformations are highlighted. In addition, Figure 2-18 highlights the expansion of the Right Ascension system to account for the motions of the coordinate systems in time and space as mentioned the introduction to this chapter. in These effects are covered in detail in Chapter 3. Table 2-3 highlights the matrix approach to the transformations amongst celestial coordinate systems. 2.4 Special Star Positions Certain positions of stars on the celestial sphere are given "special" names. As shall be seen later, some of the math models for astronomic position and azimuth determination are based on some of the special positions that stars attain. I-~-- I ECliptic If'1ea:~elestial @T o Right Ascension Mean Celestial @T Hour Angle • II Horlzon True Celestial @T CELESTIAL COORDINATE SYSTEM RELATIONSHIPS [Krakiwsky and Wells, 1971J FIGURE 2-18 Apparent @T .::. 0:> , Original System Ecliptic Ecliptic IrXl . . l:J 10 I-' Right Ascension Cf) «: iiI Horizon , RI (E:) E '"%j ...,- ::s Hour Angle Right Ascension (-~) ,/:>0 fXi l:j 1.0 R3 tLST) P2 R.A. (Jl c+ (j) a Hour Angle P2 R3 (tLST) [:J R3 (1800) Horizon TRANSFO~~TIONS R2(~-900) H.A. R2(900-~) AMONG CELESTIAL COORDINATE SYSTEMS (Krakiwsky and Wells, 1971) .• TABLE 2-3 R3 (1800) [:L, 50 Figure 2-19 shows the behavior of stars as they move in an apparent east to west path about the earth, as seen by an observer (Z) situated somewhere between the equator and the north pole (0~~S9d). Referring to Figure 2-19: 1. North Circumpolar Stars that (a) never cross the prime vertical, (b) cross the prime vertical. These stars, (a) and (b), are visible at all times to a northern observer. 2. Equatorial (a) (b) (c) 3. stars rise above and set below the celestial horizon more time above the horizon than below, equal times above and below the horizon, more times below the horizon than above. South Circumpolar Stars never rise for a northern observer. For an observer in the southern hemisphere (O~~~;:99i, of 1, 2, 3 above are reversed. the explanations Two special cases are (1) Z located on the equator, thus the observer will see 1/2 the paths of all stars, and (2) Z located at a pole, thus the observer sees all stars in that polar hemisphere as they appear to move in circles about the zenith (=po1e). 2.4.1 Rising and Setting of Stars The ability to determine the visibility of any star is fundamental to geodetic astronomy. To establish a star observing program, one must ensure that the chosen stars will be above the horizon during the desired observation period. Declination From Figure 2-20, we see that for an observer in the northern hemisphere, a north star, to be on or above the horizon - must have a declination given by o c5~90-~ , (2-48) 51 Figure 2-19 Circumpolar and Equatorial Stars 52 CELESTIAL EQUATOR I I I I I I -----------r----------"'ull .'" .. II> .. 011 .. 0 .. .. III'" "".. '. III .. '" III .. .. I._ .. .. .." .. '" .111 .. e.. .. I .. .. III.. It . .. ".." SO':ITH \J,I .... .. .... ..' 8.... 'It ... .. It .. .. .' ~ .... ...... ~ . ."f . ~' ..:'::~ ::~I~CU~POL~R.:: ': .. :. ~~ 'e. e.. e 'ST ARS ' ..... ~ It.... "" ........ • "'-v : .. ".. " I . : : .. " ~\r I· C'E.~'E.S SCP Figure 2-20 ! Declination for Visibility 53 . and a south star that will rise at some time, must have a declination (2-49) Thus, the condition for rising and setting of a star is o 0 90-IP>e>4>-90. (2-50) For example, at Fredericton, where IP= 46°N, the declination of a star must always satisfy the condition (2-50), namely in order that the star will be visible at some time; that is, the star will rise and set. If 15>44°, the star will never set (it will always be a visible circumpolar star), and if 15<-44°, the star will never rise (a south circumpolar star). Hour Angle Now that the limits for the declinations of stars for rising and setting have been defined, we must consider at what hour angle these events will occur. From the transformation of the Hour Angle to Horizon curvalinear coordinates (Section 2.3.1, (2-22» cosz = sine sin4> + cose cosh cos4> • For rising and setting,· z=90o, and the above equation reduces to sine sin IP + cose cosh coslP = 0, (2-51) which, after rearrangement of terms yields cosh = -tane tan4> • (2-52) The star's apparent motion across the celestial sphere is from east to west. (2-52): For a star that rises and sets, there are two solutions to equation the smaller solution designates setting, the larger solution designates rising (Figures 2-21, 2-22). The following example illustrates this point. Figure 2-21 . g Rising and Settl.n 0f a Star 55 Figure 2-22 Hour Angles of a Star's Rise and Set 56 At ~=46°, we wish to investigate the possible use of two stars for an observation program. Their declinations are 01=35°, 02=50°. From equation (2-50), 44°<°2< - 44°. The first star, since it satisfies (2-50), will rise and set. The second star, since 02> 44°, never sets - it is a north circumpolar star to our observer. Continuing with the first star, equation (2-52) yields = -tan35° cosh l tan46° or which is the hour angle for setting. hr 1 = The hour angle for rising is (24h_£s) = l4h 54m 05~78. 1 Azimuth At what azimuth will a star rise or set? From the sine law in the Hour Anglet:o Horizon system trans-fomtion. (equation (2-15» sinz sinA = -coso sinh. With z=900 for rising and setting, then sinA = -coso sinh. There are, of course, two solutions (Figures 2-23 and 2-24): using hr and one using h S • (2-53) one Using the previous example for star 1 (01=35°, ~=46°, h~ = 9h 05 m 54~22, h~ = 14h 54m 05~78), one gets SinA~ = -cos 35° sin 223~524l, Ar = 34° 20' 27~64. 1 Similarly AS = 325° 39' 32~36. 1 57 I I NORTH \ I POINt , Figure 2-23 Ri s~ng . and Setting of a Star 58 :Fisure 2-24 Azimuths of a Star's Rise and Set 59 The following set of rules apply for the hour angles and azimuths of rising and setting stars [Mueller, 1969]: Rise 12h<h<18h Northern Stars ·t<5>OO) Set Equatorial Stars (<5=0°) Southern Stars 270 0<A<360° Rise h=18 h A=900 Set h= 6h A=270° Rise 18h <h<24h 90 0<A<l800 Oh<h< 6h 180 0<A<2700 Set 2.4.2 6h <h<12h 00<A<900 Culmination (Transit) When a star's hour circle is coincident with the observer's celestial meridian, it is said to culminate or transit. Upper culmination (UC) is defined as being on the zenith side of the hour circle, and can occur north or south of the zenith (Figure 2-25a and 2.-25b). When UC occurs north of Z, the zenith distance is given as (Figure 25a) = <5-~ • (2-54) z = 41-<5 • (2-55) z The zenith distance of UC south of Z is Lower Culmination (LC) (Figure 2-25c) is on the nadir side of the hour circle, and for a northern observer, always occurs north of the zenith. The zenith distance at LC is z = 180- (<5+41 ) • Recalling the examples given in 2.4.1 then for the first star which is south of the zenith, and (~=46o,<51 (2-56) =35°, <5 2=50°), 60 (A) UPPER CULMINATION (8) UPPER CULMINATION .' , , I ' ,, (C) LOWER Figure 2-25 Culmination (Transit) CULMINATION 61 LC zl = 180 = 99° -(ol+~) which means that the star will not be visible (z>900). For the second star, one obtains UC z2 = 02-~ = 4° , both of which will be north of the zenith. is h = Oh The hour angle at culmination = l2h for all upper culminations, and h The azimuths at culmination are as follows: for all lower culminations. = 0° A for all Upper Culminations north of Z and all Lower Culminations; A = 180° for all Upper Culminations south of the zenith. Recalling the Hour Angle - Right Ascension coordinate transformations, we had (equation (2-30) LST = h+a • = l2h Since h = Oh for Upper Culminations and h LSTUC LC LST 2.4.3. = (l for Lower Culminations, (2-57) , = (2-58) Prime Vertical Crossing For a star to reach the prime vertical (Figure 2-26) (2-59) To compute the zenith distance of a prime vertical crossing, recall from the Horizon to Hour Angle transformation (via) the cosine rule, equation (2-26) that sino = coszsin~ + sinz cosA cos~ For a prime vertical crossing in the east, A=90° ~ltitude increasing), and for a prime vertical crossing in the west, A=270° (altitUde decreasing), 62 PRIME VERTICAL· CROSSING (WEST) Figure 2--26 Prime Vertical Crossine 63 we get 'sino =-sin\l> COSE = sin~ (2-60) cosecl%> The hour angle of a prime vertical crossing is computed as follows. From the cosine rule of the Hour Angle to Horizon transformation (equation (2-22» cosz = sino sinl%> + coso cosh cosl%> • Substituting (2-60) in the above for cosine z yields sino sinl%> = sino sinl%> + coso cosh cosl%> • or sino = sino 2 sin~ + coso cosh cos~ Now, (2-61) reduces to (2-61) sinl%> • 2 sino sin I%> coso cosl%> sinl%> sino cosh = --~~~~--~~ coso cqsl%> sinl%> or 2 cosh = tano ( cos I%> ) cosl%> sinl%> and finally cosh = (2-62) tano cotl%> As with the determination of the rising and setting of a star, there are two values for h - prime vertical eastern crossing (18h <h<24h) and prime vertical western crossing (Oh<h<6 h ). O2 = Continuing with the previous examples (01 = 35°, 50°), it is immediately evident that the second star will not cross the The first star will cross the prime vertical (0<0 1 < 46°), with azimuths A=900 (eastern) and A=270° (western). The zenith distance for both crossings will be (equation 2-60) cosz 1 = sin 35° sin 46° z 1 = 37° 07' 14~8. The hour angles of western and eastern crossings are respectively (equation 2-62) 64 W cosh2 = = tanl5 cot~ tan 35° cot 46° , ~I = 47~45397 = 3h09m48~9 and 2.4.4. Elongation When the parallactic angle (p) is 90°, that is, the hour circle and vertical circle are normal to each other, the star is said to be at elongation (Figure 1-27). Elongation can occur on both sides of the observer's celestial meridian, but only with stars that do not cross the prime vertical. Thus, the condition for elongation is that 15>!%> • (2-63) From the astronomic triangle (Figure 2-27), the sine law yields sinA sinp sin(90-15)= ~s~in~(9~0~-~!%>~) , or sinA = sinp cos!%> cosl5 (2-64) Since at elongation, p=900, (2-64) becomes sinA = cosl5 sec!%> • (2-65) For eastern elongation, it is obvious that 00<A<900, and for western elongation 270 0<A<360° •. To solve for the zenith distance and hour angle at elongation, one proceeds as follows. Using the cosine law with the astronomic triangle (Figure 1-27), one gets cos(90-!%» = cos(90-o)cosz+sinz sin(90-o)cosp , (2-66) 65 CELESl iAl I EQUATOR Figure 2-27 Elongation 66 and when p=90°, cosp=O, then sin~cosec~ cosz = (2-67) • Now, substituting the above expression for cosz (2-67) in the Hour Angle to Horizon coordinate transformation expression ,(2-22), namely = sin~ cosz sin~ + cos~ cosh cos~ , yields sin~ = sin~ sin& sin~ + cos~ sin~ - sin,~ sin~ cosh cos~ which, on rearranging terms gives cosh = 2 cosl5 sinl5 , cos~ :(2-68) Further manipulation of (2-68) leads to cosh = tan~ ( ~ ( cos 20 ) cos& sin& l-sin 26 ) = cos6sin& tan~ , and finally cosh = tan~ (2-69) cot& • Note that as with the azimuth at elongation, one will have an eastern and western value for the hour angle. Continuing with the previous examples we see that for the first star, star~ &2>~ (500~46°), &l<~' ~ ~1=35°, 02=50°, and thus it does not elongate. ~=46° - For the second thus eastern and western elongations will occur. The azimuths, zenith, distance, and hour angles for the second star are as follows: SinA~ = cos6 sec~ == cos 50° sec 46° , AE == 67° 43' 04~7 , 2 W A = 360° _AW == 292° 16' 55~3'; 2 2 cosZ z2 == sin~ cosec~ = sin 46° cosec 50°. , == 20° 06' 37~3 ; tan~ coto == tan 46° cot 50° 3• TIME SYSTEMS In the beginning of Chapter 2, it was pointed out that the position of a star on the celestial sphere, in any of the four coordinates systems, is valid for only one instant of time T. Due to many factors Ce.g. earth's motions, motions of stars), the celestial coordinates are subject to change with time. To fully understand these variations, one must be familiar with the time systems that are used. To describe time systems, there are three basic definitions that have to be stated. An epoch is a particular instant of time used to define the instant of occurance of some phenomenon or observation. A time interval is the time elapsed between two epochs, and is measured in some time scale. For civil time (the time used for everyday purposes) the units of a time scale (e.g. seconds) are considered fixed in length. time systems, the units vary in length for each system. With astronomic The adopted unit of time should be related to some repetitive physical phenomenon so that it can be easily and reliably established. The phenomenon should be free, or capable of being freed, from short period irregularities to permit interpolation and extrapolation by man-made time-keeping devices. There are three basic time systems: <.1) Sidereal and Universal Time, based on the diurnal rotation of the earth and determined by star observations, (2) Atomic Time, based on the period of electro magnetic oscillations produced by the quantum transition of the atom of Caesium 133, (3) Ephemeris Time, defined via the orbital motion of the earth about the sun. Ephemeris time is used mainly in the field of celestial mechanics and is of little interest in geodetic astronomy. 67 Sidereal and Universal 68 times are the most useful for geodetic purposes. other via ri90rouS formulae, thus the a matter of convenience. They are related to each use of one or the other is purely All broadcast time si9na1s 'are derived from Atomi time, thus the relationship between Atomic time and Sidereal or universal time is important for geodetic astronomy. 3.1 Sidereal Time The fundamental unit of the sidereal time interval is the mean sidereal day. This is defined as the interval between two successive upper transits of the mean vernal equinox (the position of T for which uniform precessional motion is accounted for and short period non-uniform nutation is removed) over some meridian (the effects of polar motion on the meridian are removed). The mean sidereal day is reckoned from Oh at upper transit, which is known as sidereal noon. 1m s = 60 ss • The units are 1h = 60m , s s Apparent (true) sidereal time (the position of T is affected by both precession and nutation), because of its variable (non-uniform) rate, is not used as a measure of time interval. Since the mean equinox is affected only be precession (nutation effects are removed), the mean sidereal day is 0~0084 shorter than the actual rotation period of the s earth. From the above definition of the fundamental unit of the sidereal time interval, we see that sidereal time is directly related to the rotation of the earth - equal an9les of an9Ular motion correspond to equal intervals of sidereal time. The sidereal epoch is numerically measured by the hour angle of the vernal equinox. The hour angle of the true vernal equinox (position of T affected by precession is termed Local Apparent Sidereal Time (LAST). and nutation) When the hour angle 69 measured is that at the Greenwich mean astronomic meridian (GHA), it is called Greenwich Apparent Sidereal Time (GAST). Note that the use of the Greenwich meridian as.a reference meridian for time systems is one of convenience and uniformity. This convenience and uniformity gives us the direct relationships between time and longitude, as well as the simplicity of publishing star coordinates that are independent of, but directly related to, the longitude of an observer. The local hour angle of the mean vernal equinox is called Local Mean Sidereal Time (LMST), and the Greenwich hour a,ng1e of the mean vernal equinox is the Greenwich Mean Sidereal Time (GMST). The difference between LAST and LMST, or equivalently, GAST and GMST, is called the Equation of the Equinoxes (Eq. E), namely Eq.E = LAST ~ LMST = GAST - GMST. (3-1) LAST, LMST, GAST, GMST, and Eq.E are all shown in Figure 3-1. The equation of the equinoxes is due to nutation and varies periodically with a maximum amplitude near 1 s (Figure 3-2). f,; It is tabulated for Oh U.T. (see 3.2) for each day of the year, in the Astromica1 Almanac (AA), formerly called the American Ephemeris and Nautical Almanac (AENA) [U.S. Naval Observatory, 1980]. To obtain the relationships between Local and Greenwich times, we require the longitude (II.) of a place. Then, from Figure 3-3, it can be seen that LMST = GMST + A ( 3-2) LAST = GAST + A , (3-3) in which A is the "reduced" astronomic longitude of the local meridian (corrections for polar motion have been made) measured east from the Greenwich meridian. 70 Figure 3-1 Sidereal Time Epochs 71 s -0.65 s -0.70 s -0.75 s -0.80 s -0.85 s -0.90 s -0.95 ~-"+---+---;---~--~--~---r---r-"-+-"-+--~--~--~ JAN. MAR. MAY JULY SEPT. Figure 3-2 Equation of Equinoxes (OhUT , 1966) [Mueller. 1969] NOV. JAN. 72 NCP ¥ TRUE Figure 3-3 Sidereal Time and Lorigitude 73 For the purpose of tabulating certain quantities with arguements of sidereal time, the concept of Greenwich Sidereal pate (G.S.D.) and Greenwich Sidereal Day Number are used. The G.S.D. is the number of mean sidereal days that have elapsed on the Greenwich mean astronomic meridian since the beginning of the sidereal day that was in progress at Greenwich noon on Jan. 1, 4713 B.C. The integral part of the G.S.D. is the Greenwich Sidereal Day Number, and the fractional part is the GMST expressed as a fraction of a day. AA shows the G.S.D. Figure 3-4, which is part of one of the tables from 74 UNIVERSAL AND SIDEREAL TIMES, 1981 Date: OhU.T, lulian Date G. SIDEREAL TIME (G. H. A. of the Equinolt) Apparent Mean Equation of Equinoxes atOhU.T. G.S.D. OhG.S.T. 244 4604.5 4605.5 4606.5 4607.5 4608.5 .h m 0 1 2 3 4 6 6 6 6 6 38 42 46 50 54 17.1886 13.7431 10.2993 06.8575 03.4174 17.9594 14.5148 11.0702 07.6255 04.1809 -0.7708 .7717 .7708 .7681 .7635 245 1299.0 1300.0 1301.0 1302.0 1303.0 5 6 7 8 9 4609.5 4610.5 4611.5 4612.5 4613.5 6 7 7 7 7 57 01 05 09 13 59.9787 56.5407 53.1023 49.6626 46.2207 . 60.7363 57.2916 53.8470 50.4024 46.9577 -0.7576 .7510 .7447 .7398 .7370 10 11 12 13 14 4614.5 461S.5 4616.5 4617.5 4618.5 7 7 7 7 7 17 21 25 29 33 42.7763 39.3297 35.8817 32.4335 28.9864 43.5131 40.0685 36.6238 33.1792 29.7346 15 16 17 18 19 4619.5 4620.5 4621.5 4622.5 4623.5 7 7 7 7 7 37 41 45 49 53 25.5415 22.0994 18.6598 15.2219 11.7843 20 21 22 23 24 4624.5 4625.5 4626.5 4627.5 4628.5 7 8 8 8 8 57 01 05 08 12 25 26 27 28 29 4629.5 4630.5 4631.5 4632.5 4633.5 8 8 8 8 8 30 31 Feb. 1 2 3 4634.5 4635.5 4636.5 4637.5 4638.5 4 5 6 7 8 Ian. I • I U.T. at OhG.M.S.T. (Greenwich Transit of the Mean Equinox) h m I 0 1 2 3 4 17 17 17 17 17 1851.3829 14 55.4734 10 59.5639 07 03.6545 0307.7450 1304.0 1305.0 1306.0 1307.0 1308.0 5 6 7 8 9 16 16 16 16 16 59 11.8356 55 15.9261 51 20.0166 4724.1072 43 28.1977 -0.7368 .7387 .7421 .7457 .7481 1309.0 1310.0 1311.0 1312.0 1313.0 10 11 12 13 14 16 16 16 16 16 39 32.2882 35 36.,3788 31 40.4693 2744.5598 23 48.6504 26.2899 22.8453 19.4006 15.9560 12.5114 -0.7484 .7459 .7409 .7341 .7271 1314.0 1315.0 1316.0 1317.0 1318.0 IS 16 17 18 19 16 16. 16 16 16 19 52.7409 IS 56.8314 1200.9220 0805.0125 04 09.1030 08.3457 04.9050 01.4618 58.0160 54.5682 09.0667 05.6221 02.1775 58.7328 55.2882 . -0.7210 .7171 .7157 .7168 .7199 1319.0 1320.0 1321.0 1322.0 1323.0 20 21 22 23 24 16 15 15 15 IS 0013.1936 56 17.2841 52 21.3746 48 25.4652 4429.5557 16 20 24 28 32 51.1192 47.6698 44.2208 40.7728 37.3264 51.8436 48.3989 44.9543 41.5097 38.0650 -0.7243 .7291 .7335 .7368 .7386 1324.0 1325.0 1326.0 1327.0 1328.0 25 26 27 28 29 15 15 15 15 15 40 33.6462 36 37.7368 3241.8273 2845.9178 24 50.0084 8 8 8 8 8 36 40 44 48 52 33.8818 30.4389 26.9975 23.5571 20.1168 34.6204 31.1758 27.7311 24.2865 20.8418 -0.7386 .7368 .7336 .7294 .7251 1329.0 1330.0 1331.0 1332.0 1333.0 30 31 Feb.' I 2 3 15 15 15 15 15 20 54.0989 1658.1894 1302.2800 09 06.3705 05 10.4610 4639.5 4640.5 4641.5 4642.5 4643.5 8 ·9 9 9 9 56 00 04 08 12 16.6755 13.2323 09.7865 06.3381 02.8879 17.3972 13.9526 10.5079 07.0633 03.6187 -0.7217 .7203 .7215 .7252 .7307 1334.0 1335.0 1336.0 1337.0 1338.0 4 5 6 7 8 15 14 14 14 14 01 57 53 49 45 9 10 11 12 13 4644.5 4645.5 4646.5 4647.5 464S.5 9 9 9 9 9 15 19 23 27 31 59.4372 55.9872 52.5392 49.0938 45.6509 60.1740 56.7294 53.2848 49.8401 46.3955 -0.7369 .7422 .7455 .7463 .7445 1339.0 1340.0 1341.0 1342.0 1343.0 9 14 41 35.0042 10 14 37 39.0948 11 14 3343.1853 12 14 2947.2758 13 14 2551.3664 14 15 4649.5 4650.5 9 35 42.2098 9 39 38.7693 42.9509 39.5062 -0.7411 -0.7369 1344.0 1345.0 14 14 21 55.4569 15 14 1759.5474 Ian. Figure 3-4. [M, 1981*] (*This and other dates in figure titles refer to year of application, not year of publication). 14.5516 18.6421 22.7326 26.8232 30.9137 75 3.2 Universal (Solar) Time The fundamental measure of the universal time interval is the mean solar day defined as the interval between two consecutive transits of a mean (fictitious) sun over a meridian. The mean sun is used in place of the true sun since one of our prerequisites for a time system is the uniformity of the associated physical phenomena. The motion of the true sun is non-uniform due to the varying velocity of the earth in its elliptical orbit about the sun and hence is not used as the physical basis for a precise timekeeping system. The mean sun is characterized by uniform sidereal motion along the equator. The right ascension (a ) m of the mean sun, which characterizes the solar motion through which mean solar time is determined, has been given by Simon Newcomb as IMueller, 1969] a + 0~0929 t 2 + ••• m m in which t m is elapsed time in Julian centuries of 36525 mean solar days which have elapsed since the standard epoch of UT of 1900 January 0.5 UT. Solar time is related to the apparent diurnal motion of the sun as seen by an observer on the earth. This motion is due in part to our motion in orbit about the sun and in part to the rotation of the earth about its polar axis. The epoch of apparent (true) solar time for any meridian is (Figure 3-5) TT in which h s = hs is the hour angle of the true sun. h convenience so that 0 (3-5) The l2h is added for TT occurs at night (lower transit) to conform with civil timekeeping practice. The epoch of mean solar time for any meridian is (Figure 3-5) MT = hm (3-6) (3-4) 76 X AT TRUE SUN· MEAN SUN Figure 3-5 Universal (5 o 1 ar ) T~me . 77 in which h m is the hour angle of the mean sun. If the hour angles of the true and mean suns are referred to Greenwich (h G s' h G ), the times are m Greenwich True Time (GTT) and Greenwich Mean Time (GMT) or Universal Time (UT) respectively (Figure 3-5). The difference between true and mean solar times at a given instant is termed the Equation of Time (Eq.T.). From Figure 3-5 and Eq.T. = TT-MT (3-7) Eq.T. = GTT (3-8) - UT. m The equation of time reaches absolute values of up to 16. Figure 3-6 shows Eq.T. for a one year interval. Mean and true solar times are related to the astronomic longitude of an observer by (Figure 3-7) MT=UT+A and TT = GTT (3-9) + A (3-10) The time required by the mean sun to make two consecutive passages over the mean vernal equinox is the tropical year. The time required by the mean sun to make a complete circuit of the equator is termed the sidereal year. These are given by 1 tropical year = 365.242 198 79 mean solar days, 1 sidereal year = 365.256 360 42 mean solar days. Since neither of the above years contain and intergral number of mean solar days, civil calendars use either a Julian 1 Juliam year d = 365.25 yea~ mean solar days in which = 365d 6 h , or the presently used Gregorian calendar which has 365.2425 mean solar days. For astronomic purposes, the astronomic year begins at Oh UT on 31 December of the previous calendar year. d astronomic date January 0.0 UT. This epoch is denoted by the 78 m -12 m --16 ~---r---r--~---r---r---r---r---r---r---r---r---r--~ Figure 3-6 Equation of Time (Oh UT, 1966) [Hueller, 1969] 79 TRUE SUN MEAN SUN Figure 3-7 Solar Ti me and Longitude so corresponding to the Greenwich Sidereal Date (GSD) is the Julian Date (JD). The"JD is the number of mean solar days that have elapsed since l2h UT on January 1 (January l~S UT) 4713 Be. d astronomic epoch of 1900 January 0.5 UT, JD For the standard = 2 415 020.0. The conversions between GSD and JD are GSD JD 3.2.1 = 0.671 = -0.669 + 1.00273790~3 (3-11) JD , (3-12) + 0.9972695664 GSD • Standard (Zone) Time To avoid the confusion of everyone keeping the mean time of their meridian, civil time is based on a "zone" concept. The standard time over a particular region of longitude corresponds to the mean time of a particular meridian. of 150 (61\.) each. In general, the world is divided into 24 zones Zone 0 has the Greenwich meridian as its "standard" o meridian, and the zone extends 7)2 on either side of Greenwich. The time zones are numbered -1, -2, •••• , -12 east, and +1, +2, ••• , +12 west. Note that zone 12 is divided into two parts, 7~°in extent each, on either side of the International Date Line (lSOoE). All of this is portrayed in Figure 3-S. Universal time is related to Zone Time (ZT) by UT = ZT + liZ , (3-13) in which liZ is the zonal correction. Care should be taken with liZ, particularly in regions where summer time or daylight saving time is used during the spring-summer-fall parts of the year. The effect is a lh advance of regular ZT. 3.3 Relationships Between Sidereal and Solar Time Epochs and Intervals We will deal first with the relationship between time epochs. 40" ••• t, • . ,.,t. ...... 20" ;'::::1 .. .... ..... ,. o· ~ (0 •• & ~ . . I-' -zo'H: : -40' i;?:: ., I: I ·co· "0; .9 100' c::J ~ ... ., ... ........ ". • :::F ~f.l\~. 'd:: •• C •• ~'::c:: •• , t t . ., ... +0 '10' :~:~: t. .... .... ~;::: I:· .. · ~ • 1 (;(;' .5 no' .4 1+3 ; +2 ao' I: :::~: ."5. *t.,. OIl. :;:t: :::E: :: t'on~t:It.O"l S +, J I) I·! Q t.ONG1Tu~t ;;.:::": f, .. "'1 I 1 ._",t' "'" .t"t_ ::::: :' .'..... .... l , .. , ... ~. ·s ,\t . ;~ It,-'t. ~ ::::::,' ...... ... -5 -K~i ~ • • t • I: -3 f -4 ....... ..... ••••••1 E ·-2 ... !.... ..... .. . 1-1 .... ... .. .. ..... .., ... :: I • " • , ~ e ..' ., :..:l~ ::\ ::1 ·12' 40 ADO 0.5 HOUR TO ZONAl. COnneCTIons AS INDICATEO. IAIlECUt.AR ZONES STANDARD (ZO~~) TIME {Mueller, 1969]. FIGURE 3-8 Note: Since the preparation of this map, the standard time zones in Canada have been changed so that all parts of the Yukon Territory now observe Pacific Standard Time (Zone designation U or +8). See Appendix II. 82 Equation (3-6) states MT = 12h + h111 . , where hm is the hour angle of the mean sun while from' Figure 3-9 (3-14) which yields (3-15) or LMST = NT h (3-16) + (am - 12 ) Equations (3-15) and (3-16) represent the transformations of LMST to MT and G vice-versa respectively. If we replace MT,hm, and LMST with GMT,nm ' and. GMST respectively, then or = and· .. GMST - (3-17) GMST= UT + (3-18) - --. For practical computations we used tabulated quantities. ..... h . - For example-c - (am -. ·12 ). is tabulated in the Astronomcal Almanac (AA) ;_. and the quantity' (am ">12h + Eq.· E) is tabulated in the Star Almanac for Land Surveyors (SALS). Thus for simplicity we should always convert MT to UT and LMST to GMST. We should note. of course, that the relationships shown (3-15), (3-16), (3-17), (3-18) relate mean solar and mean sidereal times. To relate true solar time with apparent sidereal time: (i) . compute mean solar time (MT (ii) (iii) = TT - Eq.T or OT = GTT - Eq.T.), compute mean sidereal time «3-16) or (3-18», compute apparent sidereal time (LAST = Eq.E + LMST or GAST = • Eq.E. + GMST). To relate apparent sidereal time with true solar time, the inverse procedure is used. For illustrative purposes, two numerical examples are used. , , , , 83 , , , , , , , , , , , , , , , , , , , , g~de~ea1 and un~~e~sa1 ~ime ~poc~s 'figure 3- 9 n ~e1at~ons~~Ps ~et~ee • S4 (note that the appropriate tabulated values can be found in Figures 3~lO, 3-11, 3-12, and 3-13). Example #1 (using SALS) Given: MT = ISh 21m Feb. 14, II. compute: = 4l~OO 1981 66 0 3S- 2S" W LAST MT Ish 21m 4l~OO A (_66 0 3S' 28") "'4h 26m 33~S7 R(a m - 12 22h 4Sm l4~87 II.) UT (=MT - h h + Eq.E.) at IS ~R for 4h 4Sm l4~S7: (a m - 12 h UT:9 47~4 + Eq.E.) GAST (=UT + ( am - 12h) + Eq.E) h 38 m s .39.6 , ..: :~' - ~~:. ::---.-: ~ gh 3f' 27~O 32h -2m 41~87 _24h sh GAST 11. (_66 0 38' 2S") LAST (GAST + II.) 2f1 41~87 _4h 26 m 33~S7 4h .Om 8~OO 85 Example #2 (using AA) Given: h MT = 18 21 m s 41.00 Feb. 14, 1981 A = 66° 38' 28"W Compute: . LAST MT ISh 21m 4l~00 A (-66° 3S' 2S") -4 h 26 m 33~S7 UT (=MT - A) 22h 4Sm l4~87 (a m - l2h) (GMST) at Oh UT: 9h 35m 42~95 Add Sidereal interval equivalent to (multiply UT interval by 1.002 737 9093): 22h 51 m 59~64 GMST at 22h 4Sm 14~S7 UT 32h 27m 42~59 24h GMST Eq. E GAST (= GMST + Eq. E) A (-66° 3S' 2S") LAST (= GAST + A) 8h 27m 42~59 - 0~74 Sh 27m 4l~S5 - 4h 26m 33~S7 4h 01m 07~9S 86 SUN-FEBRUARY, 1981 R U.T. ta it 0 eI 1 II 45 26•1 4625'3 47 2 4-4 84823,6 49 22 -7 12 18 0 Mon. 6 ][2 5021 •8 18 51 ... 21-0 ](8 57 IS-S 58 15- 0 859 1 4. 1 0 900 13'2 ll2 5 Thur.6 01 %2 02 %8 03 .6 0 Fri. 6 %2 II x8 II 8 0 Sun. 6 X2 %8 X2 Thur.6 4601'414 12 4600'1 13 13 18 :! X3 10•6 4556'3 u 45 5S'! II Fri. 6 x:z n·s 114554'0 10 4553'0 10 . 45 p·o 10 455 1 •0 9 II 45 50 '1 8 °3.8 II 0 8 .X X"I J2 x8 IS 0 4,5 49'3 8 454 8'5 8 4547-7 7 Sun. 6 45 47'° 6 454 6'4 6 4545. 8 6 ](6 0 Mon. 6 4545'2 .114544" 9 2 35 2 -5 245 1 '7 114542 -5 454 2 '5 45 42 -5 4542-5 so· 1ft' 40· II I> II II '" IH 4'5 4-7 4'7 4,g 4'9 S·o 5- 2 5'2 5'2 S-3 s-! S-l{ " 5-5 20· h of , 4544'7 u 4545. 1 4545'7 454 6 '3 4546'9 : . II 6' 6i 6' 7 4547·6 4548'3 7 45 49'0 ~ 4549- 8 JO· .. o· h .o',·:we " )0" II 455°'7 455 1•6 9 455 2 '5 9 4553-5'° II 45 54. 6 " h." 40· , 'Uit 50" h 55° '" 60" iii S'S 5.8 5-7 5-7 6'0 6-0 6-0 6·2 6,2 6·2, 6-3 6·S 6'3 6'3 6-4 6'4 6'7 6·6 6·S 6'9 6·8 6'7 7- 0 6'9 6-7 7. 2 7-0 6·8 7'3 7';: 7-0 "1 6·1 6'2 6'2 6'3 6-4 6·S 6·6 6'7 6·8 6'9 S'O 5.8 5'9 5'9 5'9 5'3 S-4 S'S 5-7 S'7 5'9 6·0 6,2 6-2 6,2 6-4 6'4 6'3 6·6 6-6 6'5 6·8 6·8 6'7 7'Z 7'1 7'3 7·6 7-2, 7-5 7'0 7'1 7'3 7'9 7-7 7'S F:igUrE! 3-10 [SALS, 1981] I f ! 9j 5.6 S'S 5'5 5-7 JI 45 43'S 4543,8 J 6·0 6-0 6'0 5'3 5'4 5-5 :7;6 2 3 5-7 5.8 5-3 5-4 4-6 4.8 2 2 North Latitude .10" 4-9 5.1 5-3 ;,u: I 45 44- 2 94335'3 44 34'5 4533- 6 46 32 -7 9473 1 -9 s 0 4542,6 454 2 -8 4543- 0 4543- 2 3343'9 4 1 37. 0 42 36.2 %8 0 SUNRISE 3'9 4. 2 4-4 H 0 3443- 1 South Latitude 00· . 51" II 40 37-9 %2 . I 93542 • 2 36 4 1 '3 3740 '5 38 39'.6 93938 -8 0 Sat. 6 & 45 44'7 4544'3 : 4543'9 .. 4543'5 3 45 43. 2 4543. 0 2 45 4 2 .8 1 454 2 •6 2 II 9 2 749- 1 28 48'2 2947'4 3°4 6'5 93 1 45-7 . 32 44-8 0 m 9 1 956 20 55. 1 21 54-3 . 2253'4 2550 -8 26 5°- 0 %2 45 57-5 Date 6. 0 4602 •8 IS Sun's S.D. I6!:e Feb. x X2 %8 45 58.8' II 9 1202-9 13°2 '° 14 01 •2 .15 00 '3 9 IS 59'4 ](8 24 4604',3' II '° 0 Tues.6 12'4 9 0806 '3 09 oS'S 1004-6 %2 %8 xx E It • 1757'7 IS 56-8 12 IS Cfl Cfl'2 '1 0 Sat. 6 Mon. 6 10 m 9 IS 59'4 1658.6 4609'016' Wed.6 904 09,8 05 08-9 06 08'1 IS b ~ 9 46 0'.7-1 16 4605.81 85 616 '7 . Wed. 6 46 17·8 4615'91: 4614'11 46 12-4:~ II 46 10-6 6 0 0 462 5,8 4623'721 462I '7 2O 461 9'7 20 Dec, R II eI 19 II . 85 220 - 1 Tues. 6 53 19'3 :1[2 54 18 '4 18 . 55 17-5 3. U.T. 'E II m , S 844 2 7. 0 Sun. 6 2 Dec. rn 8'2 8,1 7·8 7.6 7'4 87 INTERPOLATION TABLE :FOR R MUTUAL CO);VERSION OF INTERVALS OF SOLAR AND SIDEREAL TJ:l..JE sol31" 4b LI R sidereal J solar LJR sidereal 4b 4b 4b In solar cr;lir(l/ C'(l;I.'S LJR sidereal 5b 5h solar LJR sidereal Sb Sb muntl. Add.1R to s.olar time intt"n'..l (It"ft-hJnd argument) to oh1 .. in sidereal time inlaya!. Subtract 111< from sidereal time intcn'"l (ril(ht-h:md argument) to obtain sob. time inten'al, FIGURE 3-11 [SALS,. 1981] 88 UNIVERSAL AND SIDEREAL TIMES, 1981 Date ObU.T. Julian Date G. SIDEREAL TIME (G. H. A. of the Equinox) Apparent Mean Equation of Equinoxes atOhU.T. G. S. D. OhG.S.T. 244 4604.5 4605.5 4606.5 4607.5 4608.5 b m 0 1 2 3 4 6 6 6 6 6 38 42 46 50 54 17.1886 13.7431 10.2993 06.8575 03.4174 17.9594 14.5148 11.0702 07.6255 04.1809 -0.7708 .7717 .7708 .7681 .7635 • 245 1299.0 1300.0 1301.0 1302.0 1303.0 5 6 7 8 9 4609.s 4610.5 4611.5 4612.5 4613.5 6 7 7 7 7 .57 01 05 09 13 59.9787 56.5407 53.1023 49.6626 46.2207' 60.7363 57.2916 53.8470 50.4024 46.9577 -0.7576 .7510 .7447 .7398 .7370 10 11 12 13 14 4614.5 4615.5 4616.5 4617.5 4618.5 7 7 7 7 7 17 21 25 29 33 42.7763 39.3297 35.8817 32.4335 28.9864 43.5131 40.0685 36.6238 33.1792 29.7346 IS 17 18 19 4619.5 4620.5 462l.S 4622.5 4623.5 7 7 7 7 7 37 41 45 49 53 25.5415 22.0994 18.6598 15.2219 . 11.7843 20 21 22 23 24 4624.5 4625.5 4626.5 4627.5 4628.5 7 8 8 8 8 57 01 05 08 12 25 26 27 28 29 4629.5 4630.5 4631.5 4632.5 4633.5 8 8 8 8 8 30 31 Feb. I 2 3 4634.5 4635.5 4636.5 4637.5 4638.5 4 5 6 7 8 Jan. • • II U.T. at OhG.M.S.T. (Greenwich Transit of the Mean Equinox) h m • 0 1 2 3 4 17 17 17 17 17 1851.3829 1455.4734 10 59.5639 0703.6545 0307.7450 1304.0 1305.0 1306.0 1307.0 1308.0 S 6 7 8 9 16 16 16 16 16 59 11.8356 55 15.9261 5120.0166 47 24.1072 43 28.1977 -0.7368 .7387 .7421 .7457 .7481 1309.0 1310.0 1311.0 1312.0 1313.0 10 11 12 13 14 16 16 16 16 16 39 32.2882 3536.3788 31 40.4693 2744.s598 2348.6504 26.2899 22.8453 19.4006 t5.9560 12.5114 -0.7484 .7459 .7409 .7341 .7271 1314.0 1315.0 1316.0 1317.0 1318.0 15 16 17 18 19 16 16 16 16 16 19 52.7409 15 56.8314 12 00.9220 0805.0125 0409.1030 08.3457 04.9050 01.4618 58.0160 54.5682 09.0667 05.6221 02.1775 58.7328 55.2882 -0.7210 .7171 .7157 .7168 .7199 1319.0 1320.0 1321.0 1322.0 1323.0 20 21 22 23 24 16 IS 15 15 15 00 13.1936 56 17.2841 5221.3746 48 25.4652 4429.5557 16 20 24 28 32 51.1192 47.6698 44.2208 40.7728 37.3264 51.8436 48.3989 44.9543 41.5097 38.0650 -0.7243 .7291 .7335 .7368 .7386 1324.0 1325.0 1326.0 1327.0 1328.0 25 26 27 28 29 15 15 15 15 IS 4033.6462 3637.7368 3241.8273 2845.9178 24 50.0084 8 8 8 8 8 36 40 44 48 52 33.8818 30.4389 26.9975 23.5571 20.1168 34.6204 31.1758 27.7311 24.2865 20.8418 -0.7386 .7368 .7336 .7294 .7251 1329.0 1330.0 1331.0 1332.0 1333.0 4639.5 4640.5 4641.5 4642.5 4643.5 8 .9 9 9 9 56 00 04 08 12 16.6755 13.2323 09.7865 06.3381 02.8879 17.3972 13.9526 10.5079 07.0633 03.6187 -0.7217 .7203 .7215 .7252 .7307 . 1334.0 1335.0 1336.0 1337.0 .1338.0. 9 10 11 12 13 4644.5 4645.5 4646.5 4647.5 4648.5 9 9 9 9 9 15 19 23 27 31 59.4372 55.9872 52.5392 49.0938 45.6509 60.1740 56.7294 53.2848 49.8401 46.3955 -0.7369 .7422 .7455 .7463 .7445 1339.0 1340.0 1341.0 1342.0 1343.0 9 14 41 35.0042 10 14 3739.0948 II 14 33 43.1853 12 142947.2758 13 14 25 51.3664 14 IS 4649.5 4650.5 9 35 42.2098 9 39 38.7693 42.9509 39.5962 -0.74t1 -0.7369 1344.0 1345.0 14 14 21 55.4569 15 14 17 59.5474 16 Figure 3-12a [AA, 1981] Jan. 30 15 2054.0989 31 15 1658.1894 Feb. 15 1302.2800 2 15 09 06.3705 3 15 05 10.4610 • 4 5 6 7 8 15 14 14 14 14 0I 57 53 49 45 14.5516 18.6421 22.7326 26.8232 30.9137 89 UNIVERSAL AND SIDEREAL TIMES 1981 p G. SIDEREAL TIME (G. H. A. of the Equinox) Apparent Mean Equation of Equinoxes at O~U.T. U.T. at OhG.M.S.T. (Greenwich Transit of tbe Mean Equinox) Date OhU.T. Julian Date 244 4650.5 4651.5 4652.5 4653.5 4654.5 h m Feb. 15 16 17 .18 19 9 9 9 9 9 39 43 47 51 55 38.7693 35.3282 31.8854 28.4403 24.9927 39.5062 36.0616 32.6169 29.1723 25.7277 -0.7369 .7334 .7315 .7320 .7350 • 245 1345.0 1346.0 1347.0 1348.0 1349.0 Feb. 15 16 17 18 19 14 14 14 14 14 1759.5474 1403.6380 1007.7285 0611.8191 02 15.9096 20 21 22 23 24 4655.5 4656.5 4657.5 4658.5 4659.5 9 10 10 10 10 59 03 07 11 15 21.5429 18.0916 14.6396 11.1878 07.7369 22.2830 18.8384 15.3938 11.9491 08.5045 -0.7401 .7468 .7541 .7613 .7676 1350.0 1351.0 1352.0 1353.0 1354.0 20 21 22 23 24 13 13 13 13 13 58 20.0001 54 24.0~07 5028.1812 4632.2717 42 36.3623 25 26 27 28 Mar. 1 4660.5 4661.5 4662.5 4663.5 4664.5 10 10 10 10 10 19 23 26 30 34 04.2874 00.8396 57.3936 53.9492 50.5059 05.0599 01.6152 58.1706 54.7260 51.2813 -0.7725 .7756 .7770 .7768 .7754 1355.0 1356.0 1357.0 1358.0 1359.0 25 26 27 28 Mar. I 13 13 t3 13 13 38 40.4528 34 44.5433 30 48.6339 2652.7244 2256.8149 2 3 4 5 6 4665.5 4666.5 4667.5 4668.5 4669.5 10 10 10 10 10 38 42 46 50 54 47.0631 43.6199 40.1753 36.7284 33.2790 47.8367 44.3921 40.9474 37.5028 34.0581 -0.7735 .7721 .7721 .7743 .•7792 1360.0 1361.0 1362.0 1363.0 1364.0 '2 3 4 5 6 7 8 9 10 4670.5 4671.5 4672.5 4673.5 4674.5 10 58 11 02 11 06 11 10 11 14 29.8272 26.3744 22.9220 19.4715 16.0238 30.6135 27.1689 23.7242 20.2796 16.8350 -0.7863 .7945 .8022 .8081 .8112 . 1365.0 1366.0 1367.0 1368.0 1369.0 7 8 9 10 II 12 12 12 12 12 5921.3581 5525.4487 51 29.5392 47 33.6297 4337.7203 13 14 15 16 4675.5 4676.5 4677.5 4678.5 4679.5 11 18 12.5788 11 22 09.1358 11 26 05.6937 11 30 02.2512 11 33 58.8072 13.3903 09.9457 06.5011 03.0564 59.6118 -0.8115 .8099 .8074 .8053 .8046 1370.0 1371.0 1372.0 1373.0 1374.0 12 13 14 IS 16 12 12 12 12 12 3941.8108 35 45.9013 31 49.9919 27 54.0824 23 58.1729 17 18 19 20 21 4680.5 468t.S 4682.5 4683.5 4684.5 11 37 55.3611 11 II 11 11 41 45 49 53 56.1672 52.7225 49.2779 45.8333 42.3886 -0.8060 .8098 .8158 .8234 .8318 1375.0 1376.0 1377.0 1378.0 1379.0 17 18 19 20 21 12 12 12 12 12 2002.2635 1606.3540 1210.4445 08 14.5351 04 18.6256 22 23 24 25 26 4685.5 4686.5 4687.5 4688.5 4689.5 11 12 12 12 12 57 38.1038 01 34.6515 OS 31.2006 09 27.7515 13 24.3042 38.9440 35.4993 32.0547· 28.6101 25.1654 -0.8402 .8478 .854l .8586 .8613 1380.0 1381.0 1382.0 1383.0 1384.0 22 12 23 II 24 11 2511 26 11 0022.7161 5626.8067 5230.8972 4834.9877 4439.0783 27 28 29 30 31 4690.5 4691.5 4692.5 4693.5 4694.5 12 12 12 12 12 17 21 25 29 33 20.8585 17.4142 13.9707 10.5270 07.0825 21.7208 18.2762 14.8315 tl.3869 07.9423 -0.8623 .8619 .8609 .8599 .8597 1385.0 1386.0 1387.0 1388.0 1389.0 2711 28 It 29 II 30 II 31 11 4043.1688 3647.2593 3251.3499 2855.4404 2459.5309 1 2 4695.5 4696.5 12 37 03.6363 12 41 00.1877 04.4976 01.0530 -0.8614 -0.8653 1390.0 1391.0 11 12 Apr. • 51.9127 48.4621 45.0099 4t.S568 • G.S.D. O'G.S.T. Figure 3-12h [AA, 1981] , Apr. . m • 13· 1900.9055 13 1504.9960 13 11 09.0865 t3 07 13.1771 t3 03 17.2676 1 11 2103.6215 2 11 1707.7120 90 UNIVERSAL AND SIDEREAL TIMES. 1981 Date OhU.T. Apr. May Julian Date G. SIDEREAL TIME (G. H. A. or the Equinox) Apparent Mean • Equation or Equinoxes aIOhU.T. G.S.D. OhG.S.T. U.T. at OhG.M.S.T. (Greenwich Transit or the Mean Equinox) ... • 244 4695.5 4696.5 4697.5 4698.5 4699.5 b m 12 12 12 12 12 37 41 44 48 52 03.6363 00.1877 56.7368 53.2843 49.8316 04.4976 01.0530 57.6084 54.1637 50.7191 • 1 2 3 4 5 -0.8614 .8653 .8716 .8795 .8875 245 1390.0 1391.0 1392.0 1393.0 1394.0 6 7 8 9 10 47005 4701.5 4702.5 4703.5 47045 12 13 13 13 13 56 00 04 OS 12 46.3804 42.9320 39.4869 36.0443 32.6032 47.2745 43.8298 40.3852 36.9405 33.4959 -0.8941 .8978 .8983 .S962 .8927 1395.0 1396.0 1397.0 1398.0 1399.0 6 7 8 9 10 II 10 10 10 10 01 24.0742 5728.1647 53 32.2552 49 36.345S 45 40.4363 11 12 13 14 IS 4705.5 4706.5 4707.5 4708.5 4709.5 13 13 13 13 13 16 20 24 28 32 29.1620 25.7194 22.2748 18.8277 15.3785 30.0513 23.1620 19.7174 16.2727 -0.8893 .8872 .8872 .8896 .8942 1400.0 1401.0 1402.0 1403.0 1404.0 11 12 13 14 IS 10 10 10 10 10 41 44.5268 3748.6174. 33 52.7079 29 56.7984 2600.8890 16 17 18 19 20 4710.5 4711.5 4712.5 47135 4714.5 13 13 13 13 13 36 40 44 48 51 11.9276 08.4757 05.0237 01.5724 58.1224 12.8281 09.3835 05.9388 02.4942 59.0496 -0.9005 .9077 .9151 .9218 .9272 1405.0 1406.0 1407.0 140S.0 1409.0 16 17 18 19 20 10 10 10 10 10 22 04.9795 180.9.0700 14 13.1606 1017.2511 06 21.3416 21 22 23 24 25 4715.5 4716.5 4717.5 4718.5 4719.5 13 13 14 14 14 55 59 03 07 11 .54.6740 51.2276 47.7830 44.3399 40.8977 55.6049 52.1603 4S.7157 45.2710 41.8264 -0.9309 .9327 .9326 .9311 .9287 1410.0 1411.0 1412.0 1413.0 1414.0 21 10 02 25.4322 22 9 5829.5227 23 9 5433.6132 24 9 5037.7038 2.S 9 4641.7943 26 27 2S 29 30 4720.5 472t.S 4722.5 4723.5 4724.5 14 14 14 14 14 15 19 23 27 31 37.4556 34.0130 30.5689 27.1229 23.6746 38.3817 34.9371 31.4925 28.0478 24.6032 -0.9261 .9241 .9236 .9249 .9286 1415.0 1416.0 1417.0 1418.0 1419.0 26 27 28 29 30 1 " 5 4725.5 4726.5 4127.5 4728.5 4129.5 14 14 14 14 14 35 39 43 47 51 20.2245 16.7736 13.3235 09.8758 06.4315 21.1586 17.7139 14.2693 10.8247 07.3800 -0.9340 .9403 .9458 .9488 .9485 1420.0 1421.0 1422.0 1423.0 1424.0 6 7 8 9 10 4730.5 4731.5 4732.5 4733.5 4734.5 14 14 15 IS IS 55 58 02 06 10 02.9904 59.5516 56.1133 52.6741 49.2327 03.9354 60.490S 57.0461 53.6015 50.1568 -0.9450 .9392 .9328 .9274 .9241 1425.0 1426.0 1427.0 1428.0 1429.0 11 12 13 14 15 4735.5 4736.5 4737.5 4738.5 4739.5 IS IS IS IS IS 14 18 22 26 30 45.7888 42.3425 38.8942 35.4447 31.9949 46.7122 43.2676 39.8229 36.3783 32.9337 -0.9234 .925' .9287 .9336 .9387 1430.0 1431.0 1432.0 1433.0 1434.0 16 17 4740.5 4741.5 IS 34 28.5456 IS 38 25.0974 29.4890 26.0444 -0.9434 -0.9470 1435.0 1436.0 2 3 • ~6.6066 Figure 3-12c [AA, 1981] b Apr. May I 2 3 4 5 11 21 03.6215 11 1707.7120 II 13 11.8026 II 09 15.8931 \I OS 19.9836 9 9 9 9 9 4245.8848 3849.9754 3454.0659 3058.1564 2702.2470 1 9 2 9 3 9 4 9 S .9 2306.3375 1910.4280 15 14.5186 11 18.6091 07 22.6996 6 7 8 9 10 9 8 8 8 8 0326.7902 5930.8807 5534.9712 51 39.0618 4743.1523 12 13 14 IS " 8 8 8 8 8 4347.2428 3951.3334 3555.4239 31 59.5144 2803.6050 16 17 8 2407.6955 8 2011.7S61 91 We now turn our attention to the relationship of the sidereal and solar time intervals. The direction of the sun in space, relative to the geocentre, varies due to the earth's orbital motion about the sun. The result is that the mean solar day is longer than the mean sidereal day by approximately 4 minutes. After working out rates of change per day, one obtains , ~d (M) = 24h Ih (M) (M) , , 1m eM) , IS ld (5) = 23h 56m 04~09054 (M) lh (5) = 59m 50~17044 (M) 1m (5) = 59~83617 1s (5) = s . 0.99727 5 and in which M 03m 56~55536 (5), = 1h OOm 09~85647 (5), (M) = Olm OO~l6427 (5), (M)· = Ol~OO273 (5), refer to mean sidereal· and mean solar times respectively. These numbers are derived from the fact that a tropical (solar) year contains 366.2422 mean sidereal day~ or 365.2422 mean so~ar days. The ratio·s.are then mean solar time interval = (365.2422/366.2422) = 0.997 269 566 4x mean sidereal time interval, mean sidereal time interval = (366.2422/365.2422) = 1.002 737 909x mean solar time interval. The usefulness of a knowledge of the intervals is given in the following numerical example. 92 Example #3 (using 'M) Given: LAST = 4h OOm 04~30 Feb. 14, 1981 A = 66° 3S' 2S"W compute: ZT 4h Oom 04~30 LAST A (-66 0 38' 28") "4 h 26m 33~S7 GAST (=LAST - A) Sh 26m 3S~17 Eq.E. h (0 U.T. -{)~74 Feb. 14, 1981) Sh 26m 38~9l GMST (=GAST - Eq.E.) GMST at Oh U.T., Feb. 14, 1981· Mean Sidereal Interval (GMST 9h 3? 42:95 ii GMSTl ,,(,LU. T. U.T.(mean sidereal interval x 0.997 269 566 4) 4h ZT (= UT - ~Z) lsh 47'1'f1ll~37 93 3.4 Irregularities of Rotational Time Systems Sidereal and Universal time systems are based on the rotation of the earth; thus, they are subject to irregularities caused by (i) changes in the rotation rate of the earth, we' and (ii) changes in the position of the rotation axis (polar motion). The differences are expressed in terms of Universal time as follows: UTa is UT deduced from observations, is affected by changes in W e and polar motion, thus is an irregular time system. UTI is defined as UTa corrected for polar motion, thus it represents the true angular motion of the earth and is the system to be used for geodetic astronomy (note that it is an irregular time system due to variations in UT2 w·) • e is defined as UTI corrected for seasonal variations in w e . This time system is non-uniform because w is slowly decreasing due e to the drag of tidal forces and other reasons. UTC Universal Time Coordinated is the internationally agreed upon time system which is transmitted by most radio time stations. UTC has a defined relationship to International Atomic Time (IAT), as well as .. to UT2 and to UTI. More information on UTe, UTI, and UT2, particularly applicable to timekeeping for astronomic purposes, is given in chapter 4. 3.5 Atomic Time System Atomic time is based on the electromagnetic oscillations produced by the quantum transition of an atom. In 1967, the International committee for Weights and Measures defined the atomic second (time interval) as "the duration of 9 192 631 770 periods of radiation corresponding to 94 the transition between the two hyper-fine levels of the fundamental state of the atom of Caesium 133"jRobbins, 1976J. Atomic frequency standards are the most acurate standards in current use with stabilities of one or . 10 12 common, an d two parts ~n '1"~t~es stab~ 0 f . 10 14 a b ta~ne . d two parts ~n from. the mean of sixteen specially selected caesium standards at ·the us Naval Observatory. Various atomic time systems with different epochs are in use. The internationally accepted one is called International Atomic Time, which is based on the weighted means of atomic clock systems. throughout the world. The work required to define IAT is carried out by BIH (Bureau International de l'Heure) in Paris IRobbins, 1976]. IAT, through its relationship with UTC, is the basis for radio broadcast time signals. 4. 4.1 TIME DISSEIUNATION, TIME-KEEPING, TIME RECORDING Time Dissemination As has been noted several times already, celestial coordinate systems are subject to change with time. To make use of catalogued positions of celestial bodies for position and azimuth determination at any epoch, we require a measure of the epoch relative to the catalogued epoch. In the interests of homogeneity, it is in our best interest if on a world-wide bases everyone uses the same time scale. Time signals are broadcast by more than 30 stations around the world, with propagation frequencies in either the HF (1 to 50 MHz) or LF (10 to 100 KHz) ranges. The HF (high frequency) tend to be more useful for surveying purposes as a simple commercial radio receiver can be used. The LF (low frequency) signal are more accurate since errors due to propagation delay can be more easily accounted for. In North America, the two most commonly used time signals are those broadcast by WWV, Boulder, Colorado (2.5 MHz,S MHz, 10 MHz, 15 MHz, 20 MHz) and CHU, Ottawa (3.330 MHz, 7.335 MHz, 14.670 MHz). The broadcast signal is called Universal Time Coordinated (UTC). UTC is a system of seconds of atomic time, that is IS UTC is lSAT to the accuracy of atomic time and it is constant • h '.At 1972 January lOUT, . .UTCwas ...deUned ..to. belAT . minus . 1Us .exactly.....,w.'Ihe quantit.,y: ~.(l.AT":,,UTC), called DAT, will always be an integral number due to the introduction of the "leap second" (see below). The time we are interested in is UTI. denoted DUTI and never exceeds the value 0~9. and published by BIH. The value, (UTI-UTe) is The value DUTl is predicted Precise values are published one month in 95 arear~. 96 In addition, DUTI can be disseminated from certain time signal transmissions The code for transmission of DUTI by CHU is given in Figure 4-1. TAl is currently gaining on UTI at a rate of about one second per year. s To keep DUTI within the specified 0.9, UTe is decreased by exactly IS at certain time~. "leap second" is illustrated in Figure 4... 2. The introduction of this Note that provisions for a negative leap second are given, should it ever become necessary. All major observatories of the world transmit UTC with an error of less than O~OOI , most with errors less than O~OOO 3. The propagation delay of high frequency signals reaches values in the order of O.Sms. Since geodetic observations rarely have an accuracy of better than s 0.01 or 10 ms, an error of 0.5 ms is of little consequence. For further information on the subject of propagation delay, the reader is referred to, for example, Robbins [1976]. The various time signals consist essentially of: (i) short pulses emitted every second, the beginning of the pulse signalling the beginning of the second, ,(ii) each minute marked by some type of stress (eg. for CHU, the zero second marker of each minute is longer; for wwv, the 59th second marker is omitted)p (iii) at periodic intervals, station identification, time, and date are announced in morse code, voice, or both, (iv) DUTl is indicated by accentuation of an appropriate number of the first 15 seconds of every minute. 97 A positive value of DUT1 will be indicated by emphasizing a number (n) of consecutive seconds markers following the minute warker from seconds markers one to seconds marker (n) inclusive; (n) being an integer from 1 to 8 'inclusive. DUT1 = (n x O.l)s A negative value of DUT1 will be indicated by e~phasizing a'number (m) of consecutive seconds markers following the minute marker from seconds marker nine to seconds marker (8 + m) inclusive; (m) being an integer from 1 to 8 inclusive. DUT1 =- {m x O.l)s A zero value of DUTI will be indicated by the absence of emphasized seconds markers • The appropriate seconds markers may be emphasized, for example, by lengthening, doubling, splitting, or tone modulation of the normal seconds markers. EXAMPLES DUTI MI NUTE MARKER ,___ = + 0.55 EMPHASIZED SECONDS MARKERS ----A. . .---..,., o I DUTI = - 0.25 MINUTE M'ARKER EMPHASIZED SECONDS MARKERS r-A-. LIMIT OF CODED SEQUENCE Figure 4-1 Code for the Transmission of DUT1 I I I --I I 98 A positive or negative leap second, when required, should be the last second of a UTe month, but;,preierenceshould be,' given to the end of December and June and second preference to the end of March and September. 1(,. positive leap second begins at 23 h 59,m. 60 S a~d ends.~tOh Om O\of~he ~ir~t day of' the following month~ In the case of anegat1.ve leap second, 23 59 58 w:Lll be followed one second later by Oh Om Os of the first. day of the. following month .. . . Taking account of what h~s been said in t~e preceding paragraph, the dating of events in the vicinity of a leap second shall be effected in the manner indicated in the following figures: POSITIVE LEAP SECOND EVENT DESIGNATION OF THE DATE OF THE EVENT t ~lEAP SECOND I 56 I 51 58 I 59 I :I 60 I 0 30 JUNE, 23h59m ~ I 1 2 I 3 ~h . m s . 30 JUNE, 23 59 60.6 UTe 4s 1 JULY OhOm NEGATIVE LEAP SECOND DESIGNATION OF THE DATE OF THE EVENT EVENT I 56 I .I 57 58 30 JUNE, 23h59m t a0 1I ~I~ I 2 1 JULY I I 4 3 5 I t m s 30 JUNE, 23 59 58.9 6 ohom Fir.;ure 4-2 Dating of Events in the Vicinity of A Leap Second UTC 99 4.2 Time~Keeping and Recording Time reception, before the advent of radio, was by means of the telegraph. Since the 1930's, the radio receiver has been used almost exclusively in geodetic astronomy. Time signals are received and amplified, after which they are used as a means of determining the exact time of an observation. Direct comparison of a time signal with the instant of observation is not made in practice - a clock (chronometer) is used as an intermediate means of timekeeping. Usually, several comparisons of clock time with UTe are made during periods when direction observations are not taking place. The direction obs~rvations are subsequently compared to clock time. The most practical receiver to use is a commercial portable radio with a shortwave (HF) band. In areas where reception is difficult, a marine or aeronautical radio with a good antenna may become necessary. The chronometer or clock used should be one that is very stable, that is, it should have a constant rate. There are three broad categories of clock that may be used: (1) (2) (3) Atomic clocks, Quartz crystal clock, Mechanical clock. Atomic clocks are primarily laboratory instruments, are very costly, and have an accuracy that is superfluous for geodetic purposes. Quartz crystal clocks are the preferred clocks. They are portable, stable (accuracy of 1 part in 10 7 to 1 part in 10 12 ), but do require a source of electricity. This type of time-keeper is normally used for geodetic purposes, in conjunction with a time-recorder, (e.g. chronograph) where an accuracy of O~OOI or better is required. Mechanical clocks, with 100 accuracies in the order of 1 part in 10 6 , suffice for many surveying needs (eg. where an accuracy of O~l is required). They are subject to mechanical failure and sudden variations in rate, but' they are portable and independent of an outside source of electricity. The most common and useful mechanical clock is a stop watch, reading to O~l, with two sweep seconds hands. One should note that many of the new watches - mechanical, electronic, quartz crystal - combine the portable-stable criteria and the stop watch capabilities, and should not be overlooked. With any of these latter devices, time-recording is manual, and the observing procedure is audio-visual. 4.3 Time Observations In many instances, the set of observations made for the astronomic determination of position and azimuth includes the accurate determination and recording of time. While in many cases,(latitudeand azimuth mathematical models and associated star observing programs can be devised to minimize the effects of random and systematic errors in time, the errors in time always have a direct effect on longitude (e.g. if time is s in error by 0.1, the error in A will be 1~'5). The timekeeping device that is most effective for general surveying purposes is the stop watch, most often a stop watch with a mean solar rate (sidereal rate chronometers are available). The five desireable elements for the stop watch are lRobbins, 1976]: (i) (ii) two sweep seconds hands, a start-stop-reset button which operates both hands together, resetting them to zero, 101 (iii) a stop-reset button which operates on the secondary seconds hand only, resetting it to coincidence with the main hand which is unaffected by the operation of this button, (iv) (v) reading to 0~1, a rate sufficiently uniform over at least one half an hour to ensure that linear interpolation over this period does not s give an error greater than 0.1. A recommended time observing procedure is as follows [Robbins, 1976]: (i) start the watch close to a whole minute of time as given by the radio time signal, (ii) stop the secondary hand on a whole time signal second, record the reading and reset. A mean of five readings of time differences (see below) will give the 6T of the stop watch at the time of comparison. (iii) stop the secondary hand the instant the celestral body is in the desired position in the telescope (this is done by the observer). (iv) (v) Record the reading and reset. repeat (iii) as direction observations are made. m repeat (ii) periodically (at least every 30 ) so that the clock rate for each time observation· can be determined. The clock correction (6T) and clock rate (6 1T) are important parts of the determination of time for astronomic position and azimuth determination. Designating the time read on our watch as T, then the zone time (broadcast, for example, by CHU) of an observation can be stated as ZT =·T + t:.T (4-1) 102 In (4-ll! it is assumed that ~T is constant. ~T will usually not be constant but changes should occur.at a uniform rate, bolT. Then boT is given by (4-2) where ~T,~T ~lT o are clock corrections at times T, T 0 is the change in ~T , per unit of time, the clock (chronometer) rate. The clock correction, ~T for any time of observation, is determined via a simple linear interpolation as follows: (i) using the radio and clock comparisons made before, during, and after the observing program, compute ~T, 1 (ii) ~T for each by = ZT, - T, 1 (4-3) 1 Compute the time difference, boZT, '+1 between time comparisons 1,1 by ~ZTi,i+l = ZT i +l - ZT i (iii) Compute the differences between successive clock corrections corresponding to the differences boT,1.,1.+ , 1= (iv) ~T '+11. ~ZT, '+1 ' namely 1,1. boT,1 (4-5) Compute the clock rate per mean solar hour by ~ tv) (4-4) 1 T = ~T,1,1.'+1 ~ZT,1.,1'+1 Record the results in a convenient and usable form (e.g. table of values, graph). (4-6) 103 This time determination process is illustrated by the following example. Clock Correction and Clock Rate (i) Determination of ZT of stop watch ZT (radio) Stop Watch (T) ~T, 1 gh 13m OO~O Oh OOm OO~O gh 13m OO~O 13m 15~0 OOm 14~8 gh 13m 00~2 13m 45~0 OOm 45~2 gh 12m 59~8 gh 14m OO~O Oh Olm OO~O gh 13m OO~O 14m 30~0 Olm 29~9 gh 13m 00~1 (Radio-Stop watch) gh 13m oo~b i=l 1 This procedure is to be repeated for each time comparison with ZT of beginning = (i~T,)/i = the radio time signal. (ii) Record of time comparisons; determination of clock rate. ZT (radio) ~ Clock (stop watch) ~ZT, '+1 1,1 TIME (T) ~T gh13mOO~O ohOomOO~O gh13mOO~0 gh41mOO~0 Oh27m59~2 10h30mOO~0 Ih16m57~7 llh33mOO~0 2h19m55~7 Oh28mOO~0 Oh49mOO~0 Ih03mOO~0 0 gh13mOO~8 gh13m02~3 gh13m04~3 T ~T.1,1'+1 (~/h) 00~8 1.7 01~5 1.8 02~0 1.g 104 Graph of Clock (stop watch) Correction ~T (iii) --t:i ."..,- • CLOCK TIME (To) (iv) Determination of the true zone time of a direction observation. Observed Time (stop watch): What is true ZT ? From the graph: ~T at ZT observed = 9 28~8 + ZT (true) = Oh S2 m = 10h Using ~T 6T = ~To = gh + (T i +1 - Ti)~lT 13m ZT(true) OO~8 + = Oh S2 m OSm gh 13m 01~6 30~4 yields (O~40803) = 10h OSm h 13m 01.6 s 28~8 + 30~3 • 1.8 = gh 13m gh 13m 01~6 Ol~S 105 In closing this section, the reader is cautioned that the methods described here for time determination are not adequate for precise (e.g. first-order) determinations of astronomic azimuth and position. s work requires an accuracy in time of 0.01 or better. For details, the reader is referred to, for example, Mueller I1969J or Robbins 11976]. Such 5. 5.1 STAR CATALOGUES AND EPHEMERIDES Star CataloguesStar catalo9Ues contain the listing of the positions of stars in a unique coordinate system. The positions are generally given in the mean right ascension coordinate system for a particular epoch T. o Besides listing the right ascensions and declinations, star catalogues identify each star by number and/or name, and should give their proper motions. Some catalogues also include annual and secular variations. Other pertinent data sometimes given are estimates of the standard errors of the coordinates and their variations, and star magnitudes. (Magnitude is an estimate of a star's brightness given on a numerical scale varying from -2 for a bright star to +15 for a dim star). There are two main types of catalogues: (i) Observation catalogues containing the results of particular observing programs, and (ii) compilation catalogues containing data from a selection of catalogues (observation and/or other compilation catalogues). The latter group are of interest to us for position and azimuth determination. Several compilation catalogues contain accurate star coordinates for a welldistributed (about the celestial sphere) selection of stars. These catalogues are called fundamental catalogues and the star coordinates contained therein define a fundamental coordinate system. Three of these catalogues are described briefly below. The Fourth Fundamental Catalogue (FK4) IFricke and Kopff, 1963] was produced by the Astronomischen Rechen Institute of Heidelberg, Germany in 1963. It was compiled from 158 different observation catalogues and contains coordinates for 1535 stars for the epochs 1950.0 and 1975.0. 106 107 A supplement to :the FK4, with star coordinates for 1950.0, contains additional stars. catalogue. .!2!!1 These additions were drawn mainly form the N30 It should be noted that the FK4 system ha's been accepted internationally as being the best available, and all star coordinates should be expressed in this system. The General or Boss General Catalogue (Ge, or BC) [Boss, 1937), compiled from two hundred fifty observation catalogues, contains star coordinates for 33 342 stars for the epoch 1950.0. The present accuracy of the coordinates in this catalogue are somewhat doubtful. GC-FK4 correction tables are available, but are of doubtful value due to the errors in the GC coordinates. The Normal System N30 (N30) IMorgan, 1952] is a catalogue of 5268 Standard Stars with coordinates for the epoch 1950.0. Although more accurate than the Ge, it is not of FK4 quality. For geodetic purposes, the most useful catalogue is the Apparent Places of Fundamental Stars (APFS) IAstronomische RechenInstitutes, Heidelberg, 1979]. This is an annual volume containing the apparent places of the FK4 stars, tabulated at 10-day intervals. As the name implies, the published coordinates have not been corrected for short period nutation terms that must be applied before the coordinates are used for position and azimuth determination. For all but latitude observations there are sufficient stars in this publication. For the stringent star- pairing required for latitude determinations there are often insufficient stars to fulfill a first-order observation program. In such cases, the FK4 supplement should be referred to and the coordinates rigorously updated from 1950.0 to the date of observation. 108 5.2 . E;phemexoides and Almanacs Ephemerides and l\lmanacs are annual volumes containing info:rmation of interest to surveyors and others (e.g. mariners) involved with "practical" astronomy. They contain the coordinates of selected stars, tabulated for short intervals of time, and may also contain some or all of the following information: motions of the sun, moon, and planets of our solar system, eclipses, occultations, tides, times of sunrises and sunsets, astronomic refraction, conversions of time and angular measures. The Astronomical Almanac (AA) [U.S. Naval Observatory, Nautical Almanac . , , Office, 1980]. ,(formerly,. published ,j;l1.thelJittted" States -as·'the· American Ephemeris and Nautical Almanac and in 'the United Kingdom as the Astronomical Ephemeris) is the only one of geodeticinte;-e~t:.· It'contai;lJs~amongst' other things~ the mean I?lac~s of 1475 s'tars- catalogued for the "beginning of theye'ar' of'irtteres"t, (e~ g~ 1981.0), plus the information required to update the star coordinates to the time of observation. All of the information given is fully explained in a section at the end of each AA. Parts of three tables are given here - Figures 5-1, 5-2, 5-3, - that are of interest to us. Each table is self- explanatory since the info:rmationhas been previously explained. note concerning Figure 5-1 is as follows. GMST = UT + (a m - An extra Recall from equation (3-18) that l2 h ). Now, since the tabulated values are for ohUT, and ST is the hour angle of the vernal equinox, then GAST at OhUT = (a at OhUT = (a GMST m - l2h + Eq. E), m For more detailed information, the interested reader should study a recent copy of AA. 109 UNIVERSAL AND SIDEREAL TIMES. 1981 Date OhU.T. Jan. Julian Date G. SIDEREAL TIME (G. H. A. of the Equinox) Apparent Mean Equation of Equinoxes atOhU.T. G.S.D. OhG.S.T. U.T. at OhG.M.S.T. (Greenwich Transit of the ~fean Equinox) • 17.9594 14.5148 11.0702 07.6255 04.1809 -0.7708 .7717 .7708 .7681 .7635 • 245 1299.0 1300.0 1301.0 1302.0 1303.0 60.7363 57.2916 53.8470 50.4024 46.9577 -0.7576 .7510 .7447 .7398 .7370 1304.0 1305.0 1306.0 1307.0 1308.0 42.7763 39.3297 35.8817 32.4335 28.9864 43.5131 40.0685 36.6238 33.1792 29.7346 -0.7368 .7387 .7421 .7457 .7481 1309.0 1310.0 1311.0 1312.0 1313.0 10 11 12 t3 14 37 41 45 49 53 25.5415 22.0994 18.6598 15.2219 11.7843 26.2899 22.8453 19.4006 15.9560 12.5114 -0.7484 .7459 .7409 .7341 .7271 1314.0 1315.0 1316.0 1317.0 1318.0 16 17 18 19 IS 16 16 16 16 16 1.9 52.7409 15 56.8314 1200.9220 08 05.0125 04 09.1030 57 01 05 08 12 08.3457 04.9050 01.4618 58.0160 54.5682 09.0667 05.6221 02.1775 58.7328 55.2882 -0.7210 .7171 .7157 .7168 .7199 1319.0 1320.0 1321.0 1322.0 1323.0 20 21 22 23 24 16 15 15 IS IS 00 56 52 48 ·44 4629.5 4630.5 4631.5 4632.5 4633.5 8 16 51.1192 8 20 47.6698 8 24 44.2208 8 28 40.7728 8 32 37.3264 51.8436 48.3989 44.9543 41.5097 38.0650 -0.7243 .7291 .7335 .7368 .7386 1324.0 1325.0 1326.0 1327.0 1328.0 25 26 27 28 29 15 15 15 15 15 40 33.6462 3637.7368 3241.8273 2845.9178 2450.0084 2 3 4634.5 4635.5 4636.5 4637.5 4638.5 8 8 8 8 8 33.8818 30.4389 26.9975 23.5571 20.1168 34.6204 31.1758 27.7311 24.2865 20.8418 -0.7386 .7368 .7336 .7294 .7251 1329.0 1330.0 1331.0 1332.0 1333.0 4 5 6 7 8 4639.5 4640.5 4641.5 4642.5 4643.5 8 56 16.6755 9 00 J3.2323 9 04 09.7865 9 08 06.3381 9 12'02.8879 17.3972 13.9526 10.5079 07.0633 03.6187 -0.7217 .7203 .7215 .7252 .7307 1334.0 1335.0 1336.0 1337.0 1338.0 9 4644.5 4645.5 4646.5 4647.5 4648.5 9 15 59.4372 10 11 12 13 55.9872 52.5392 49.0938 45.6509 60.1740 56.7294 53.2848 49.8401 46.3955 -0.7369 .7422 .7455 .7463 .7445 1339.0 1340.0 1341.0 1342.0 1343.0 9 10 11 12 13 14 15 4649.5 4650.5 9 35 42.2098 9 39 38.7693 42.9509 39.5062 -0.7411 -0.7369 1344.0 1345.0 14 14 21 55.4569 15 14 17 59.5474 244 4604.5 4605.5 4606.5 4607.5 4608.5 .11 III 0 1 2 3 4 6 6 6 6 6 38 42 46 50 54 5 6 7 8 9 4609.5 4610.5 4611.5 4612.5 4613.5 6 7 7 7 7 57 01 05 09 10 11 12 13 14 4614.5 4615.5 4616.5 4617.5 4618.5 7 7 7 7 7 17 21 25 29 33 15 16 17 18 19 4619.5 4620.5 4621.5 4622.5 4623.5 ·7 7 7 7 7 20 21 22 23 24 4624.5 4625.5 . 4626.5 4627.5 4628.5 7 8 8 8 8 25 26 27 28 29 30 31 • 17.1886 13.7431 10.2993 06.8575 03.4174 59.9787 56.5407 53.1023 49.6626 13 46.2207· I II Jan. 0 1 2 3 4 17 17 17 17 17 m 1851.3829 1455.4734 1059.5639 07 03.6545 0307.7450 5 16 59 11.8356 6 16 55 15.9261 7 16 51 20.0166 8 16 47 24.1072 9 16 4328.1977 / Feb. I 9 9 9 9 36 40 44 48 52 19 23 27 31 Figure 5-1 [AA, 1981] 16 16 16 16 16 39 35 31 27 23 32.2882 36.3788 40.4693 44.5598 48.6504 13.1936 17.2841 21.3746 25.4652 29.5557 30 15 2054.0989 31 15 1658.1894 Feb. I IS 1302.2800 2 15 09 06.3705 3 15 05 10.4610 4 15 14 6 14 7 14 8 14 01 14.5516 57 18.6421 53 22.7326 '4926.8232 45 30.9137 14 14 14 14 14 4135.004·2 37 39.0948 3343.1853 2947.2758 2551.3664 .s 110 BRIGHT STARS, 1981.0 Name B.S. Right Ascension h B Ocl • . .. Declination Notes 0 V B-V +1.27 +1.63 -0.05 +1.04 -0.11 Spectral IS 38.2 59.1 46.0 21.7 24.1 -77 10 14 - 607 11 -17 26 30 -54850 +28 59 08 F,V F,V F,V F F 4.78 4.41 4.55 4.61 2.06 11 fJ Cas E Phe 22 And B Sci 88 y Peg 21 25 27 35 39 008 o OS 009 o 10 o 12 09.3 27.0 19.6 46.1 15.3 +59 -45 +45 -35 +15 F.S F F F F,S,V 2.27 +0.34 3.S8 +1.03 5.03 +0.40 5.25 +0.44 2.83 -0.23 89 " Peg 7 Cet 2S fT And 8 , Cet t Tuc 45 48 68 74 77 37.0 40.5 19.8 27.5 OS.3 +200604 -1902 17 +36 40 4S - 8 55 45 -64 59 11 F,S 4.80 4.44 4.52 3.S6 4.23 +1.57 +1.66 +0.05 +1.22 +0.58 41 Psc 27 p And fJ Hyi Ie Phe a Phe 80 82 98 100 99 o 13 o 13 o 17 o 18 o 19 o 19 o 20 o 24 o 25 + 805 +37 51 -77 21 -43 47 -4224 F F F 5.37 5.18 2.80 3.94 2.39 +1.34 +0.42 +0.62 +0.17 +1.09 gK3 F6 Gl A5 KO IV IV Vn IlIb AI Phc fJlTuc IS Ie Cas 29 'II' And 118 125 126 130 154 o 29 o 30 25.7 -23 53 34 F 30.1 -48 54 30 F 030 40.8 -63 03 46 o 31 54.5 +62 49 38 F,S o 35 51.7 +33 3654 F 5.19 4.77 4.37 4.16 4.36 +0.12 +0.02 -0.07 +0.14 -0.14 A5 AO B9 Bl BS Vn V V la V 153 157 163 165 168 o 35 o 36 54.3 +53 47 33 19.8 +35 17 43 037 32.8 +29 12 32 o 38 18.5 +3045 26 o 39 25.2 +562600 F S F F,S F,V 3.66 -0.20 5.48 +0.88 4.37 +0.87 3.27 +1.28 2.23 +1.17 B2 IV G3 11 G8 IlIp K3 III KO- IlIa 16 fJ Cel 22 0 Cas 34 And 180 191 188 193 21S 040 042 042 043 046 25.8 -46 11 21 F 30.2 -57 34 02 F,D 38.1 -1805 27 F 39.6 +48 10 50 F 19.7 +24 09 51 F,V 4.59 4.36 2.04 4.54 4.06 G8 III B9 Vp Kl JJl B5 III KIll 63 8 A 24 .., 64 35 " Psc Hyi Cas Psc And 224 236 219 225 226 19 4I"Cct Psc Cet Psc 33 21 a And 17 t Cas 30 E And 31 8 And 18 a Cas ". Phe '1 Phc , 3 37.1 06.9 46.4 16.2 025 20.S o 47 02 51 58 14 04 +0.97 +0.00 +1.02 -0.07 +1.12 -104447 +64 OS 40 - I 14 50 -69 37 47 +6036 51 F F,C F F F,V 5.I9 5.39 4.77 5.45 2.47 +38 +23 -29 + 7 -46 F 3.87 +0.13 4.42 +0.94 4.31 -0.16 4.28 +0.96 3.31 +0.89 20 Cct A"Tuc 27 y Cas 235 233 248 270 264 049 049 o 52 o 54 055 37 ". And 38 .., And a Sci 71 E Psc fJ Phe 269 271 280 294 322 o 55 o 56 o 57 41.6 11.3 41.4 I 01 57.3 I OS 14.2 F 4.43 +1.50 5.07 +1.37 3.44 +0.57 5.07 +0.51 4.53 -0.15 + 728 -75 01 +57 42 +16 50 +40 58 10.4 33.6 02.1 18.0 33.0 OS 49 41 07 33 F F F F F S F F 41.7 56.1 047 56.5 047 58.6 04845.6 o 47 42 08 00 22 41 23 18 27 47 49 Figure 5-2 55 36 55 18 33 48 56 36 17 13 F.S F 2 [AA, 1981] +0.50 +0.49 +1.57 +1.09 -0.15 Type K2 III M3 III B9 IV Kl III B9p 000 000 002 004 007 30 2 9OS4 90S9 9098 DI F2 III-IV KO III F2 II dF4 B21V M2 III MI 111 A2 V Kl.S III F9 V K5 K4 GO F8 B5 III III V V V F8 V gGO + A5 MO- Ula K2 III BO.S IYel A5 G8 B7 KO G8 V Ill-IV III (C III III II) 111 BRIGHT STARS, 1981.0 Nama B.S. Right Ascension II In I .," Declination Notes V B-V 4.25 5.37 5.21 5.17 3.92 +1.21 +0.88 +0.16 +0.69 -0.08 K2I11 05 III -A3 IV/V 05 Vp 'B7 V Spectral , Tuc II Phe 30 p. Cas , Phe 285 332 331 321 338 1 1 1 1 1 05 54.1 06 33.7 06 55.8 0659.8 07 35.3 +8609 -61 52 -41 35 +54 49 -55 20 31 42 43 33 84 'IJ Cet <I> And fJ And 8 Cas X Psc: 334 335 337 343 351 I 1 1 I 1 07 08 08 09 10 -10 1658 F +47 08 27 +35 31 13 F +55 02 57 +205602 F 3.45 4.25 2.06 4.33 4.66 +1.16 -0.07 +1.58 +0.17 +1.03 Kl III 87 III MO lila A7 V 08 III 83 T Psc: 86 , Psc: Ie Tuc: 89 Psc: 90 " Psc: 352 361 377 378 383 1 10 36.6 1 1244.2 I 15 07.6 1 1649.0 1 18 25.1 +295921 + 728 31 -68 58 37 + 3 3053 +270953 F F F F 4.51 4.86 4.86 5.16 4.76 +1.09 +0.32 +0.47 +0.07 +0.03 KO FO F6 A3 A2 III·IV Vo IV V V 34 46 45 37 36 Cas And Cet Cas Cas 382 390 402 403 399 1 1 1 1 1 18 21 23 24 24 52.5 12.9 04.3 33.6 34.4 +5807 56 +45 25 47 - 8 1652 +6008 13 +68 01 53 S,M F F F.S,V F 4.98 +0.68 4.88 +1.08 3.60 +1.06 2.68 +0.13 4.74 +1.05 FO KO KO AS KO Ia III·IV IlIb III·IV II1 Psc: And 'Y Phe 48 Cet 3 Phe 414 417 429 433 440 I 1 I I 1 25 26 27 28 30 39.9 30.7 32.5 41.4 27.7 +1908 +45 18 -43 24 -21 43 -49 10 33 33 55 38 16 F F F F.7 F 5.50 4.83 3.41 5.12 3.95 +1.11 +0.42 +1.57 +0.02 +0.99 99 'I Psc: 50 " And 51 And 40 Cas a Eri 437 458 464 456 472 1 1 1 1 1 30 35 36 36 37 27.8 40.5 49.1 58.2 00.5 +15 +41 +48 +72 -57 1454 18 39 31 57 56 38 19 59 F,3 F F F,D F 3.62 4.09 3.57 5.28 0.46 +0.97 +0.54 +1.28 +0.96 -0.16 G8 FS K3 08 83 111 V III It·III Vp 489 490 497 500 496 1 1 1 1 1 40 40 41 41 42 26.4 57.3 17.1 45.8 27.7 + 5 23 +35 09 -32 25 - 347 +5035 F F +1.36 -0.09 +1.04 +1.38 -0.04 K3 B9 gKO K3 B2 III IV·V F,M F F,V 4.44 5.40 5.26 4.99 4.07 S3 X Cet 509 510 514 513 531 1 1 1 1 I 43 44 44 45 48 11.1 23.3 45.3 01.9 39.0 -1602 14 + 903 45 -25 08 50 - 54941 -104648 F F,S F S F 3.50 4.26 5.31 5.34 4.67 +0.72 +0.96 +0.39 +1.52 +0.33 G8 Vp G8 III dFI K4 III FlV 55 Cet 2 a Tri III ( Psc: til Phe 45 « Cas 539 544 549 555 542 I I 1 1 1 50 51 52 52 53 31.3 59.6 34.2 53.1 00.8 -102543 +2929 13 + 3 05 39 -4623 43 +63 34 38 F F F F F 3.73 3.41 4.62 4.41 3.38 +1.14 +0.49 +0.94 +1.S9 -0.15 K2 111 F6 IV KO J1l M4III B3 Vp <I>,Phe 6 IJ ,Ari 1)' Hyi )( Eri o Hyi 558 553 570 566 1 53 34.7 -42 35 24 I 53 35.2 +2042 56 I 54 27.1 -67 44 26 I 5S 13.1 -SI 42 II 1 58 10.3 -61 3943 F F F F F S.1l 2.64 4.69 3.70 2.86 -0.06 +0.13 Ap +0.95 G8.S 1lI +0.85 +0.28 G8 nIb FO V 94 48 106 <I> ( 8 3 til Co> I' Psc: f'( Scl <I> Per 52 llO T 0 E Cet Psc ScI , S91 38.0 23.5 39.8 56.2 25.7 22 36 18 40 50 31 01 21 08 37 F F F,D F V Figure 5-3 [AA, 1981} Type SKI FS V K5+ lIb-lIla AI V KO lIIb II·III Ve4p A5 V CN·2 112 The Star Almanac for Land Surveyor§ (SALS) rH.M. Nautical Almanac Office, 1_980], published annually, is expressly designed to meet the surveyor's requirements in astronomy. (i) The main tables included are: the Sun, in which the right ascension, declination, and equation of time are tabulated for 6-hourly intervals (see explanations below regarding Figure 5-4), (ji) the Stars, in which the apparent right ascension and declination of 685 stars are given for the beginning of each ~onth (iii) h (e.g. 0 UT, March 1, 1981), Northern and Southern Circumpolar Stars (five of each), are listed separately, their apparent right ascensions and declinations being given at lO-day intervals for the year, (iv) the Pole Star Table, a special table for Polaris (n Ursae Minoris) for use in specific math models for latitude and azimuth determinations. Also included in SALS are several supplementary tables, some of which are companions to those noted above (for interpolation purposes). Examples of three SALS tables are given in Figures 5-4, 5-5, and 5-6. Most of the information given has been explained previously, and the tables are self explanatory. Regarding the terms R and E in Figure 5-4, one should note that h R= (n - 12 m E + Eq.E) = GAST at ohUT, = Eq.T. - 12h • The reader is cautioned that SALS is not suitable for use where first-order standards mus-t be met.<AA <may be used, but common practice is to use one of the fundamental catalogues and compute the updated star coordinates via procedures outlined in Chapter 6. 113 SUN-FEBRUARY, 1981 U.T. U.T•. d 1 d " 0 12 12 ' 18 18 12 :12 18 . 18 0 2 35 2 '5 10 0 Tues. 6 3. 0 Tues. 6 II 9 Wed, 6 12 :12 :18 18 1:Z 0 0 Wed. 6 Thur,6 12 12 18 18 5 0 Thur.6 Fri. 6 :12 12 18 18 6 IJ 0 14 0 0 Sat. 6 Fri. 6 7 12 12 18 18 IS 0 Sun. 6 12 12 18 18 16 0 Mon. 6 12 12 18 i8 24 24 Sun"s S.D, Feb. 5S· " 4'4" ~ II ~ .. ~ II 2 2 3 4543'5 4543,8 4544'2 4544'7 II 4545. 1 4545'7 4546'3 4546'9 II 45 47·6 4548 '3 45 49'0 4549.8 II 3 4 5 4: 6 6 6 7 7 ~ 9 45 50'7 455 1•6 9 455 2'5 9 4553'S 10 114554.611 II North Latitude ~ ~ ~ 5'3 5'4 5'5 5'5 5'5 5'7 5'7 5'7 5.8 5'9 5'9 5'9 6'0 6·0 6'0 6,2 6'2 6,2 6'3 6'3 6'3 6'5 6'4 6'4 6'7 6·6 6'5 6'9 6·8 6'7 7'0 6'9 6'7 7,2 7'0 6·8 7'3 7·2 7·0 7·6 7'4 7,1 5'7 5'7 5'9 6'0 6,1 6·% 6,2 6'3 6'4 6'5 6·6 6'7 6-8 6'9 [SALS, 1981J II W 5'% S'J Figure 5-4 " ~ 8,% 8'1 7,8 5'5 • ¢ 7'9 7'7 7'5 5'4 II ~ • 5'3 110 ~ II :z6 • ~ 7'3 7·6 7. 2 7'5 7-I' 7'3 31 II ~ 7'2 7·1 7'0 5-1 5'3 5'4 II ~ 6·8 6·8 6'7 4'9 5'1 5-3 " ~ 6·6 6,6 6'5 4.6 4.8 5'0 21 2 6'4 6'4 6'J 5.1 5,2 16 I 6,2 6·% 6'2 4'5 4-7 6 0 0 6·0 6·0 6'0 5,2 II 0 5'7 5.8 5,8 4'9 3'9 4'2 4'4 4542 '5 4542'5 4542'5 II 45 42 ,6 4542,8 4543'0 4543. 2 5'5 5'5 5.6 4'7 4.8 5,0 :I I 93938.8 4°37'9 41 37'0 42 36.2 South Latitude 60- • 4544'7 ' 4544'3 : 4543'9 4 4543'5 3 45 43'2 4543.0 2 454 2 •8 2 4542•6 2 SUNRISE 16=2 Date m I 1 454 2 'S 94335'3 44 34'5 4533.6 46 32 '7 9473 1 '9 0 Sun. 6 II 245 1 '7 2550,8 26 5°'° 9 2749. 1 28 48'2 2947'4 3°46 '5 ,93 1 45'7 32 44.8 3343'9 3443. 1 93542 • 2 36 41 '3 3740'5 38 39.6 0 Sat. 6 8 II 9 1956,0 20 55'1 21 54'3 2253'4 0 Mon. 6 ... .. 9 0 Mon. 6 Sun. 6 2 E Dec. R " II 114 RIGHT ASCENSION OF STARS, 1981 No. Mag. R.A. Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Jan. • • • • 5' 52 53 54 55 4-3 4'4 4-3 4-0 4'3 221 2. 26 22.7 238 2. 39 25-0 17'3 oS-6 30 -3 18'5 23-3 16·6 oS'3 29'9 16·8 21'7 15.8 07. 8 29'5 15. 2 20·6 15'3 07'5 29.2 14'0 20·2 15'2 07·6 29' 1 13'5 • 20·8 22·2 24.2 15.6 16'5 17'7 oS-o oS·8 09'7 29-5 30'3 31. 2 13'9 15,'2 17.0 26,1 27-5 28'0 27·6 18·8 19'7 20'1 20·0 10·6 11'3 II-7 u·8 32.1 32.8 33'2 33'3 19'0 20'5 21'1 20·8 56 57 58 S9 4'1 3. 6 4. 2 4'4 4'4 2. 2. 39 42 2. 42 2. 43 243 54'4 IS'4 53'4 12'5 54'3 51'4 17'9 54'0 29'5 32 16'7 32.8 24·6 57'0 55'5 53.8 53'3 18'0 17-6 52'7 .52'1 II'7 12'1 53'9 53. 6 53'2 17·6 52.1 II'7 53-5 53-5 IS,O 52.6 12-0 53-9 55'4 19-7 55'0 13'7 55. 6 56 '5 20·6 56 '3 14·6 56'5 57'3 21·2 57·2 15'3 57- 2 50·8 17'0 53. 1 29. 0 31 '3 16'3 32'3 23. 8 56'4 54. 8 S0· 5 16'3 52 '5 28·6 30'7 50'4 16,2 52 '4 2S·6 30 -5 50'9 16·S 52 '9 28'9 30 ;8 15.8 31'7 22'9 55. 8 54. 2 16'2 32 ~3'4 53'7 20'9 56.8 31'4 33'7 18'7 34. 6 27'3 59-4 57. 8 54'4 22·0 57-8 32'1 15'9 31.8 23. 1 55'9 54'3 'P'2 15-5 4°- 0 09-9 57·6 40'4 14'9 39'5 09'2 56 -8 39.8 14'4 39'0 oS'5 56 . 1 40'1 14'5 39.1 oS'5 56 '3 46'4 47'1 07'3 oS·o 29'5 30'7 48'4 . 49'0 01'0 01·6 52'7 19'5 55'4 30'5 32.6 17.8 33. 6 25'9· 58'3 56 '7 42'5 16,1 40·6 10'3 58 .6 48.0 oS'9 32'3 50·0 02'5 II 60 61 62 63 64 65 66 6'] 68 84 85 86 I I 57'7 21'7 57'9 15'7 57'7 55-0 22·8 58.6 32.6 35'0 19'9 35.8 29'2 60'9 59'4 45·6 18'4 4 2'9 t 3.0 61·8 I 3.8 3 23 3 26 3 27 3 29 332' 47'4 oS-4 3 2 '2 49'5 02·2 47-1 oS·o 31 '4 49- 1 01·8 46 -6 07·6 300 4 48'7 01'3 46'2 07'2 29'4 48.2 00'9 332. 57'1 54'3 341 34. 6 342 20'4 3 43 07·8 56'7 53'9 34'1 20'1 07-4 56 . 1 55.6 53'5 53. 0 33'3 32'7 19.6 19.1 06·8 06'3 55'4 52'9 32'4 18'9 06,1 55. 6 56'2 53. 1 53'7 32'7 33.6 19'1 19'7 06'4 07'1 57. 1 54.6 34.8 20·6 oS'l 43. 8 17.1 41.6 II'4 59'9 48'9 49.6 50·2 09. 8 10·6 11'1 33'9 35'3 36 '3 50'9 51'7 52'2 03'3 04'1 04·6 58 '1 58.8 59'4 55'5 56.2 56.8 36. I 37'2 38 '0 21'5 22'2 22·8 09'2 10-1 10·8 343 343 343 346 346 44'9 54'4 59.1 02·1 21'4 44·6 54'0 57. 8 01'7 21·1 44. 1 53-3 56'3 01·1 20·6 43.6 52'7 55'0 00·6 20·1 43'4 52'4 54. 1 00'4 19'9 43'7 52'7 54'1 00'5 20'1 45'3 54'7 56'2 01'9 21·8 46 '3 55. 8 58 02'9 22'7 347 348 34f8 352 3 56 34'4 02-1 45- 0 56'4 35-0 3 57 357 3 58 3 59 402 given oS·8 44.1 28,1 37.8 oS-9 refer 25'9 25.6 26·6 28-6 31. 1 33'5 34'9 35. 1 00'5 00·8 01'5 02'4 03'4 04. 2 04'9 05-3 42'9 43'0 43·6 44-5 45'5 46-4 47'0 47'3 54. 8 55'0 55'7 56'7 57'7 58-7 59'4 59'9 33'1 33'3 34'0 35. 1 36 '3 37'3 38 '1 38.6 oS'5 oS·o 07'5 07- 2 07'4 07'9 oS-7 09.6 10'4 II'I II-4 43.8 43.2 42.6 42'4 42.6 43'3 44'3 45'4 46'4 47'2 47-7 27'0 25'7 24'4 23·6 23'5 24- 1 25'4 26'9 28· 3 29' 3 29.6 37' 5 37'0 36 .6 36 '4 36 '5 37·1 38 38-9 39'7 400 4 40.8 oS·6 oS'I 07'7 07'4 -07·6 oS'2 09- 0 09'9 10'7 1I'4 II·8 to the beginning of the month. and should be interpolated to actual date by means of the table on page 73- 3'7 4'4 4'3 3- 8 4'3 . 4'4 3. 1 3'7 3'9 90 3'2 3·8 4'2 2'9 3'0 ¢ 3'2 97 4'0 C)8 4'4 99 3'9 100 3-9 The figures 301 3 01 3 03 3 03 3 06 33S '° 32'2 01'7 44'5 56.1 34.6 29'7 01·2 43.8 55-5 33-9 '° 56'2 54. 6 54'3 18·8 53'7 12·8 54'7 51'7 18'0 54'0 29.6 31.6 16:9 32'7 24'5 57'1 55'5 4 1-2 15'2 39-7 09'2 ,57'3 44'4 53'5 54.8 01'1 20·8 27'4 00-7 43'2 55.0 33'3 Figure 5-5a ,° [SALS, 1981] ,° 34'S 19'4 35'3 28'4 60'3 58'7 44-9 17'9 42-3 12'4 61'0 47·1 56-9 59' 5 03'7 23.6 47·8 57'7 60'5 04'3 24'3 • 26'4 19.6 II·6 33. 2 '19'7 57'7 57'4 21·8 21'7 58. I 57. 8 15. 8 15'7 57'9 57-8 55. 2 55- I 23'0 22'7 58.8 58.6 32'7 32 •6. 35. 1 34.8 20·1 20·0 36'0 35.8 29'5 29'3 61'2 61·1 59'7 59. 6 46•0 45.8 18·6 18'4 43'1 43'° 13-2 12'9 62·2 62-1 39'7 14'2 38.8 oS'3 55'9 46. 1 07'0 29'1 48.1 00'7 B9 93 94 95 s 3 u: 3 18 3 19 3 Z% 88 9Z • 30'] S, 91 • 4'2 3'9 3-9 4'3 1'9 ' 2·8 69 81 82 83 • 55'0 IS·8 54. 1 12'9 54'7 51'S IS,S 54-7 29. 8 32 •6 17·1 33'3 25'4 57'5 56-0 41.8 15'9 40'4 10'5 5S-2 70 71 72 73 74 75 76 77 78 79 80 m 50-5 II'4 50 '5 II'4 36 '9 52.6 04'9 36'7 52 •6 04'9 59'7 59. 6 57- I 57'1 38'5 38'5 23.1 23'1 II·2 II'3 48 '2 48'3 58. 2 58 '2 60-7 60,1 °4. 6 04. 6 24-7 24'7 33'9 °5'4 47'2 59'9 38'7 II'S 47.8 29.2 4°'9 11'9 the 115 DECLINATION OF STARS, 1981 Name No. Dec. Jan. F. M. Apr. M. J. July A. 8 51 52 S3 S4 5S S6 57 58 61 6% 63 64 65· 66 67 68 69 70 '11 '12 '13 74 7S 76 77 78 79 80 81 82 83 84 85 86 S, 88 89 90 91 92 93 94 95 96 97 98' 99 100 • II II S4746 102104101 N 822 25 23 22 N 014 40 38 37 S6820 78 80 76 S 39 5S 91 94 92 N 309 14 12 II N49 08 61 62 59 ,S 1356 33 3S 3S N 1001 58 56 55 N2'] 10 57 57 55 N SS..s 69 70 68 N 5%40 73 75 72 S 858 36 38 39 S 40 2.Z 69 73 71 N4 00 49 47 46 S2341: 70 73 73 N 5325 63 65 63 N 3845 64 65 63 N40S2 62 63 61 N4932 34 35 34 S29 03 58 62 61 S21: 49 50 54 54 S43 08 50 54 53 N4947 43 45 44 N 857 39 37'36 N 939 55 54 52 N 595% 35 38 38 N 12,52 14 13 12 S 931: 31 34 35 S 21 41 60 64 65 N020 22 20 19 N4743 42 44 44 S949 49 52 53 N 32 x3 43 43 43 N24 03 13 13 12 N 4231 II 13 12 S645X SI 86 85 S23 x8 34 38 39 N24 02 46 46 45 S 7417 73 77 76 N 2359 43 43 42 S J6 IS 43 48 49 N 31 49 39 40 39 N3957 22 24 23 S x333 55 59 60 N3S44 13 14 14 S6126 94 99 99 N x226 08 C1] 06 N 5S6 C1] 05 04 Figure 5-5b S. Oct. N. D. Jan. " 29 8" " ," 27 22" 22" 95 85 75 65 60 22 23 26 31 36 ~2 8 38 41 45 51 56 68 58 46 37 31 E 86 78 68 59 52 Eridani L II 14 17 Ceti 23 28 'Y e Persei 54 49 45 44 46 Ceti 'IT 33 28 21 14 08 Ceti 54' 55 58 62 68 j.l. 52 50 50 52' 56 41 Arietis Persei 63 57 52 50 51 TJ 68 62 57 55 57 Persei T Eridani 37 33 27 20 14 TJ 66 58 48 39 32 e Eridani' (It Ceti 46 48 51 56 62 T3 Eridani 70 64 56 47 41 59 53 48 46 47 'Y Persei 60 56 53 53 56 p Persei Algol (13 Persei) 57 53 50 50 52 L Persei 30 24 20 18 20 (It Fornacis 57 SO 42 33 26 16 Eridani 51 45 38 30 23 BS 1008 (Eri.) 48 40 30 20 13 (It Persei 40 35 31 29 30 0 Tauri 36 37 39 43 48 52 53 55 59 64 ~ Tauri BS 1035 (Cam.) 33 27 21 17 17 II 12 13 Tauri 17 21 5 Eridani E 33 29 23 17 II Til Eridani 62 56 49 41 34 10 Tauri 19 21 25 31 36 & Persei 40 36 31 29 30 52 48 42 35 29 & Eridani 0 Persei 40 38 36 36 38 10 (1)08 10 13 17 Tauri 00 01 (1) 05 02 v Persei 80 71 60 49 41 13 Reticuli TI Eridani 36 30 23 15 08 44 43 42 44 47 TJ Tauri 71 61 50 40 32 'Y Hydn 27 Tauri 40 39 39 40 43 BS 1195 (En.) 45 38 29 19 12 37 3S 33 33 3S t Persei 21 17 14 13 14 Persei E 58 54 48 41 35 'Y Eridani 12 (1) 06 06C1] E Persei & Reticuli 94 86 75 64 56 ). 060608 II 15 Tauri V Tauri 04 05 08 12 17 No., mag., dist. and p.a. of companion star: 65. 4'4. 8". 880 K 59 60 Hydri Eridani Ceti Ceti Hydri • II II S6844 70" 71 66" 58 47 36 G [SALS, 1981] 59 41 61 31 50 32 52 04 72 .60 56 6% II II 63 43 62 37 54 34 S8 05 75 65 63 69 10 32 29 66 68 38 39 5% 58 60 66 57 62 24 30 23 24 19 ,20 10 13 33 39 52 55 68 71 21 26 25 28 3 71 44 61 46 60 34 65 08 76 69 71 76 13 39 68 43 65 71 68 37 30 24 20 45 55 71 34 29 07 06 09 30 40 33 %5 42 17 31 41 38 24 46 20 04 39 04 50 30 47 08 39 17 30 10 53 19 21 35 40 43 27 50 22 09 14 42 50 04 09 54 56 33 42 50 52 ,09 IS 42 21 30 14 55 21 23 46 25 33 IS 63 22 22 47 79 43 S9 5S 53 85 42 57 62 68 75 30 32 72 12 75 72 79, 83 17 47 77 17 74 73 84 88 20 54 66 64 50 55 73 78 76 79 73 76 43 48 37 43 30 36 28 35 52 57 54 ' 53 70 69 42 49 29, 28 13 17 41 47 38 35 50 55 31 36 53 ' 55 24, 25 19 23 60 68 15 21 58 59 52 60 54 55 23 31 49 51 30 34 38 43 22 25 73 82 21 20 20 18 116 POLE STAR TABLE, 1981 Jd' L.S.T. m o 3 6 9 12 IS 18 21 24 27 30 33 J6 39 42 45 ',.s 51 54 57 60 , , , , Bo , . . , , -26'5 -4 1 ,6 -14.8 -47'0 - 2·2 -49'2 +10·6 -48,° +22'7 -43'5 +33'2 -36'1 14'2 47.2 23'2 43. 2 26'0 42'0 1'5 II'3 47·8 33.6 35'7 49'2 13·6 47'4 25'4 42 '3 u'9 47'7 23. 8 42 '9 34'1 35'3 °'9 49'2 24,8 42'7 13.0 47·6 - 0·2 49. 2 24'4 4 2.6 34'5 ,34.8 12'5 47'5 24'3 43'0 12'3 47'7 + 0'4 49. 2 13'1 47'3 24'9 42 '2 35'0 34'4 -23'7 -43'3 -I1'7 -47'9 + 1·1 -49'2 +13'7 -47,1 +25'5 -4 1 '9 +35'4 -33'9 II·I 48'0 23'2 43.6 1'7 49'2 14'4 46'9 26'0 41.6 35'9 33'4 22,6 43'9 10'5 48'2 2'4 49.1 15'0 46'7 26·6 41'2 36'3 33'° 22-0 44,2 2']-1 40'9 9,8 48'3 3'0 49. 1 ,15.6 46'5 36 •8 32 '5 2'].6 40 -5 16'2 46'3 37'2 32•0 3.6 49'0 21'4 44'5 9'2 48'4 -20·8 -44'7 - 8·6 -48'5 7'9 48,6 20'3 45'0 19'7 45'3 7'3 48'7 6·6 48.8 19. 1 45'S 6,0 48'9 18'5 45.8 + 4'3 -49'0 +16·8 -46'1 +28-2 -40,2 +37·6 -3 1 '5 4'9 5.6 6'2 6·8 48'9 48.8 48.8 48'7 17'4 18'0 18·6 19'2 45'9 45'7 45'4 45'2 28'7 29'2 29'7 30'2 39.8 39'4 39'0 38.6 38•0 38 '4 38.8 39'2 31'0 3°·6 3°,1 29'5 -17'9 -46,° - 5'4 -49'0 ,+ 7'S -48.6 +19.8 -44'9 +3°-7 -38 '2 +39.6 -29- 0 8,1 48'5 17'3 46.2 20'4 44·6 31 '2 37·8 39'9 28'5 4'7 49'0 4. 1 49'1 40'3 28,0 16'7 46'4 20'9 44'4 31 -7 37'4 8'7 48'3 16'1 46.6 21'5 44.1 32 •2 37- 0 40'7 27-S 9'4 48 .2 3'4 49'1 22·1 43.8 32'7 36.6 2·8 49'1 10·0 48.1 15'4 46.8 41 26'9 -14.8 -47'0 - 2·2 -49'2 +10·6 -48,° +22-7 -43'5 +33'2 -36 '1 +4 1 '4 -26'4 '° Lat. o o 10 20 30 40 45 50 $5 60 62 64 66 Month Jan. -'3 -'3 -·2 -·2 ,-·1 -·1 ·0 +·1 +,2 +·2 +·2 +'3 '0 -'1 -'1 -·2 -'2 -'3 -'4 -'3 -'3 -'4 -'4 -'3 -·2 -·2 -·1 -'4 -'3 -'3 -'1 -'3 -'3 -'2 -'2 -·2 -·2 -·2 -'1 -·1 -I '0 -·1 '0 -·1 -·1 -'1 ·0 -·1 ,0 ·0 ·0 '0 oQ +·1 ·0 ·0 ·0 '0 +·1 +'1 -'1 -'1 +'2 +·2 +'3 +'4 +·1 +.J -·1 -·1 ·0 ·0 '0 +·1 +'2 +'2 +'3 +'4 -·1 -·1 B2 +·1 +-2 +'1 +'2 +'2 +'3, +'2 +'3 +'2 b2 B2 b2 '0 '0 '0 -'4 -'3 -'2 -'2 -'3 -'2 -·1 -·1 -'1 -·1 -·1 ·0 ·0 ,0 '0 +.J ,0 +·1 -'4 -'4 -'3 -'2 -'1 -·1 +'2 +'2 +'3 +'3 +'2 +'2 +'3 +'4 B2 hI bl +2 . +-2 +'3 -·2 -·2 -'3 -'1 -·2 --I . -·1 ·0 -·1 -'3 -·1 -'3 ·0 +-1 +·1 '0 Feb. Mar. +·1 -·1 ·0 ·0 ,0 '0 -0 '0 ·0 -.J '0 '0 Apr. JuJy Aug. -'4 -'5 -·1 '-'5 -'5 -'4 +·1 +,2, +'2 +.J -.J '0 +·1 +·1 +·1 -'1 -'2 June ·0 -·1 -'2 -'3 -'5 -'S -·S +,2 +·2 +'3 +'2 -'3 -'5 -'4 May Sept. Oct. Nov. Dec. ·0 -·1 -·1 -'3 -'3 ·0 -·1 -'3 -'4 -'3 -'4 -'3 -'3 -'4 -'5 -·6 -'4 -'3 -·6 +'2 +'4 -·2 -'2 -·6 '0 -·6 -'5 +'2 +'4 +'5 -'4 '0 -·1 '0 -·2 -'3 -'5 -·6 -·6 -'4 -'4 -'3 --I -·2 -'4 -'5 -·6 -'4 -·1 -'2 -'3 -.6 -·2 --6 -·1 -·6 +-1 -·6 -'5 Latitude = Corrected observed altitude of Polaris + a. + a, + 82 +'1 +·2 -·1 Azimuth of Polaris = (bo + b l + b2 l sec (latitude) Figure 5-6 [SALS, 1981] -0 ·0 -·1 -'3 ·0 -·2 -'4 -'S -'S -'3 --S -'4 -·6 -·s -'2 6. VARIATIONS IN CELESTIAL COORDINATES The mathematical models for the determination of astronomic latitude, longitude, and azimuth require the use of a celestial bodies apparent (true) coordinates for the epoch of observation. The coordinates (a,S) of the stars and the sun (the most often observed celestial bodies for surveying purposes) are given in star catalogues, ephemerides, and almanacs for certain predicted epochs that do not (except in exceptional circumstances) coincide with the epoch of observation. requirements stated previously, the coordinates (a,~) To fulfill the must be updated. As was indicated in Chapter 5, the updating procedure has been made very simple for users of annual publications such as ,".,,~.j SALS, and APFS. The question is, how are the coordinates in the annual publications derived, and how should one proceed when using one of the fundamental star catalogues (eg. FK4). The aim of this chapter is to answer this question. In previous discussions in these notes, coordinates in the Right Ascension system have been considered constant with respect to time, and in the Hour Angle and Horizon systems, changes occurred only as a result of earth rotation. In this chapter, we consider the following motions in the context of the Right Ascension system: (i) precession and nutation: motions of the coordinate system relative to the stars; (ii) proper motion: relative motion of the stars with respect to each other; (iii) refraction, aberation, parallax: apparent displacement of stars due to physical phenomena; 117 118 (iv) polar motion: motion of the coordinate system with respect to the solid earth. Except for polar motion, the above factors are discussed in terms of their effects on a and 0 of a celestial body. in the Right Ascension coordinate system. This leads to variations The definition of the variations are given below and Figure 6-1 outlines their interrelationships. (i) Mean Place: heliocentric, referred to some specified mean equator and equinox. (ii) True Place: heliocentric, referred to the true equator and equinox of date when the celestial body is actually observed. (iii) Apparent Place: geocentric, referred to the true equator and equinox of date when the celestial body is actually observed. (iv) Observed Place: topocentric, as determined by means of direct readings on some instrument corrected for systematic instrumental errors (e.g. dislevellment, collimation). The definitions are amplified in context in the following sections. 6.1 Precession, Nutation, and Proper Motion The resultant of the attractive forces of the sun, moon, and other planets on our non-symmetrical, non-homogeneous earth, causes a moment which tries to rotate the equatorial plane into the ecliptic plane. We view this as the motion, of the earth's axis of rotation about the ecliptic pole. This motion is referred to as precession. The predominant effect of luni-solar precession is a westerly motion of the equinox along the equator of 50~3 per year. approximately 26000 years, and its is approximately 23~5. The period of the motion is amplitude (obliquity of the ecliptic) Superimposed on this planetary precession, which 119 MEAN @ To . ) I PROPER MOTION PRECESSION ( I MEAN @ T I NUTATION I @ T TRUE I ANNUAL 'ABERATION ANNUAL PARALLAX I APPARENT @ T I DIURNAL ABERATION GEOCENTRIC PARALLAX REFRACTION J OBSERVED @ T Figure 6,-1 Variations of the Right Ascension Syste~ 120 causes a westerly motion of the equinox of l2~5 per century, and a change in the obliquity of the ecliptic of 47" per century. General precession is then the sum of luni-so1ar and planetary precession. Within the long period precession is the shorter period astronomic nutation. The latter is the result of the earth's motion about the sun, the moon about the earth, and the moon's orbit not lying in the ecliptic plane. The period of this motion is about 19 years, with an ampli tude of 9". General precession, with astronomic nutation superimposed, is shown in Figure 6-2. Each star appears to have a small motion of its own, designated as its proper motion. This motion is the resultant of the actual motion of the star in space and of its. apparent motion due to the changing direction arising from the motion of the sun. In Chapter 5, it was stated that certain epochs T have been o chosen as standard epochs to which tabulated mean celestial coordinates (a ,0 ) of celestial bodies refer. o 0 The mean celestial system is completely defined by: a heliocentric origin, a primary (Z) pole that is precessing (not nutating), and is called the mean celestial pole, a primary axis (X) that is precessing (not nutating), following the motion of the mean vernal equinox, a Y-axis that makes the system right-handed. Now, to define a set of coordinates (a,c) for an epoch T, we must update from T to T, since for every epoch T, a different mean celestial system o 121 GENERAL PRECESSION =NUTATION . AT NT o T To o PATH OF TRUE CELESTIAL POLE P = PRECESSION I\J T = NUTATION AT T PATH OF MEAN CELESTIAL POLE Figure 6-2 Motion of the Celestial Pole 122 is defined. The relationship between mean celestial systems is defined in terms of precessional elements (~ o ,e, z) (Figure 6-3) and proper motion a d elements (po,po). Expressions for the precessional elements were derived by Simon Newcomb around 1900, and are [e.g. Mueller, 1969] z; o z = = (2304~1250 + 1~396t ) t + 0~302t2 + 0~018t3 o z; + 0~179lt2 + 0~100lt3 (6-2) o e = (2004~1682 - 0~853t ) t - 0~1426t2 - 0~042t3 o in which the initial epoch is T = 1900.0 + t o T = 1900.0 + t o + t, in which t (6-1) 0 0 (6-3) , and the final epoch and t are measured in tropical centuries. The precessional elements L~o' z,e) are tabulated fo;r,tp.~;b~gi1'lp.i~go;f.the current year in AA. From Figure 6-3, it can be seen that x y X = (6-4) y z z 'Me T o or, setting (6-5) then x x y z =P (6-6) y z Proper motion of a star is accounted for in the mean right ascension system, and its effects on (a ,0 ) are part of the update o 0 from a mean place at To to a mean place at T. here without proof) is The transformation (given 123 NCP (T) ~----.~YMCT Figure 6-3 Mean Celestial Coordinate Syste~s 124 x = y (6-7) in which t =T - T and dV are the annual tangential component of dt proper motion and rate of change of that component, and W is the direction o (in years), ~ o (azimuth) of proper motion at T. o a Catalogue such as the FK4 are ~o The quantities tabulated in a Fundamental 0 ' Vo ' the annual components of proper motion in right ascension and declination, and their rates of change per a one hundred years, d Vo / dt, cS d~o/dt. The.~nual~, d~/dt, and Wo are then given by d a ~o dV/dt dt - + = , a ( ~o (080 0 ) = cos -1 V Setting 1 2 (6-8) (6-7) is rewritten as x y X =M (6-9) y z MCT o Now, combining (6-6) and (6-9), the complete update from a mean place at T o to a mean place at T is given by 125' x x = PM Y z (6-10) Y Z The effects of astronomic nutation, which must be accounted for to update from the mean celestial system at T to the true celestial system at T (Figure 6-1), are expressed as nutation in longitude nutation in the obliquity (6E). Expressions for 6~ (6~) and and 6E have been developed, and for our purposes, values are tabulated (e.g. AA). The true celestial system at T to which (ao'oo) are updated is defined by: a heliocentric origin, - a primary pole (Z) that is the true celestial pole following the precessing and nutating axis of rotation, - a primary axis ~) that is the true precessing and nutating vernal equinox, - a Y-axis that makes the system right-handed. The true and mean celestial systems are shown in Figure 6-4, and from this, we can deduce the transformation x x = Y Z TC T Z Y (6-11) .Designating N = ~(-E-6E)R3(-6~)R1(E) , (6-12) ZTC A NCP\JMEAN) "NCP (TRUE) ~ ~c YTC ECLIPTIC PLANE Figure 6-4 True and Uean Celestial Coordinate Systems ...... N 0\ 127 (6-11) is written x x Y = N Z Combining (6-13) Y Z (6~6), (6~9), and (6~11), the update from mean celestial at T o to true celestial at T is given in one expression, namely x x Y = NPM Z Z 6.2 Y Annual Aberration and Parallax The apparent place celestial system is the one in which the astronomic coordinates (41, h) are expressed, thus it is in this system that the mathematical models (relating observables, known and unknown quantities) are formulated. This requires that the (no'oo) of a celestial body be updated to the apparent place system, defined as having: - an origin coincident with the earth's centre of gravity, ~ a primary pole (Z) coincident with the earth's instantaneous rotation axis, ~ a primary axis (X) coincident with the instantaneous vernal equinox, a Y-axis that makes the system right-handed. Evidently, the main change is a shift in origin. This gives rise to two physical effects (i) annual parallax due to the shift in origin, and (ii) annual aberration due to the revolution of the new origin about the heliocentre. 128 Aberration is the apparent displacement of a celestial body caused by the finite velocity of light propagation combined with the relative motion of the observer and the body. The aberration we are concerned with here is a subset of planetary aberration, namely annual aberration. general law of aberration depicted in Figure 6~5, From the and our knowledge of the earth's motion about the he1iocentre, the changes in coordinates due to annual aberration CflaA,M A) are given by Ie.g. Mueller, 1969] =- K flo A = .... K fla In A (6~l5) and seco (cosa cosA cose s + sina sin>.. . s ), (6-15) (cosh . s cose [tane coso· ..... s·ina sin·h . s ). (6~16), A is the ecliptic longitude of the sun, a,o are in s the true celestial system at T, and K is the constant of aberration for the earth's orbit (assumed circular for our purposes), given numerically as ~ = 20~4958. In matrix form, the change in the position vector is given by ...-D c A= C (6-16) tans in which C and D are referred to as the aberrational (Besselian) 'day numbers (Note: C = -K cose cos>.. s J D = -K sinA s ). These quantities are tabulated, for example in M: Parallactic displacement is defined as the angle between the directions of a celestial Object as seen from an observer and from some standard point of reference. Annual (Stellar) Parallax occurs due to the separation of the earth and sun, and is the difference between a geocentric direction and a heliocentric direction to a celestial body (Figure 6-6). Expressed as changes to (a,o) we have 129 ~ ____________=A~5----------_.51 ~ O~ S- ~ ~ i'Q' 9' ~l<; R 9." r.:>~ ~ ~ $. g,(,j o ~ S: POSITION OF STAR AT LIGHT EMISSION 0: POSITION OF OBSERVER AT LIGHT DETECTION DIREC~ION OF OBSERVER MOTION VI VELOCITY OF OBSERVER MOTION S5: PARALLEL TO 00' SUCH THAT 5SI _ V_ SO-C- K od: WHERE' C VELOCITY OF LIGHT IN A VACUUM, AND K CONSTANT OF ABERRATION = ----------~-----------·I o Figure 6-5 Aberration 130 I I \ \ I \ , I " "'" -- ...... '" 9 : HELIOCENTRIC DIRECTION 9' : GEOCENTRIC DIRECTION :II : ANNUAL (STELLAR) PARALLAX Figure 6-6 Annual (Stellar) Parallax 131 flo. p = flo = II p II (6",,17) {COSo. COSE sin}, . s ..... sino. cos). s ) seco (coso sinE sinA. .. coso. sino COSA. s s .. sino. sino COSE sinA. ), (6-18) s in which the new quantity II is the annual (stellar) p!irallax. quantity - for the nearest star it is effects of aberration, 0.,0 in {6-17} system at T. 0~76. II is a small As for the computation of the and (6-18) are in the true celestial Using the Besselian Day Numbers C and D, the changes in the position vector are given by R= -c seCE -D COSE -D sinE (II/K) To update (0.,0) from the True Celestial system at T to the Apparent Celestial system at T, we write the expression x x y = A + R + z y (6-20) z Finally, the complete update from Mean Celestial system at T to Apparent o Celestial at T is given by the expression x y z x =A + R + NPM A.P· T (6-21) y z MCT o This entire process is summarised in Figure 6-7, and all symbols used in Figure 6-7 are explained in Table 6-1. 132 CATALOGUED ~He1iocentric) ao'tS o ' (To) costS cosa o 0 costS sina o 0 sintS o .' - = CATALOGUED XO' Yo' Z0 I PROPER MOTION -, M = R3 (-ao)R2(tSo-90)R3(~o)R2(~t+1/2~~t2) R3 (-~o)R2(90-tSo)R3(ao) PRECESSION NUTATION I l ANNUAL ABERRATIONl ANNUAL PARALLAXt A = ntanE] [ CseCE ]~/k R = - Dcose: Dsine: X APPARENT (X, Y, Z) APPARENT a' a', tS~ =R Y Z (Geocentric) X + A + NPM Y 0 0 Zo = tan -1 y/x = sin- 1 Z = tan- 1 z/(x2+y2)1/2 (T) FIGURE 6-7 POSITION UPDATING M.e. To to A.P. T 133 Description of Terms Symbol o0 ) (a , 0 Description Where Obtained Right ascension and declination of the star at epoch T From Fundamental Catalogue Direction numbers of star at T From (a , 0 0 (X , Y ,Z ) 000 0 ) 0 0 II Annual Tangential component of proper motion of star at T • 0 Catalogue Rate of change of proper motion t Time interval in years t=T-T ljJo Direction (azimuth) of proper motion at T From Fundamental Catalogue Precessional Elements From Astronomical z, e) (6ljJ, 6£) £ (C,D) II K (X,Y, Z) ! Fundamental dll dt 0 (1;;0' From "1 '1 (a,o ) Nutational Elements Obliquity of the ecliptic Aberrational Day Numbers 0 Ephemeris From Astronomical Ephemeris Stellar Parallax of Star From Fundamental Catalogue Constant of annual Aberration K=20".4958 Direction numbers of the star at epoch T Computed from all the above parameters Right ascension and declination of the star at epoch T, referred to the Apparent (Geocentric) System. TABLE 6-1 Computed from (X,Y,Z) at time T 134 6.3 Diurnal Aberration, Geocentric Parallax, and Astronomic Refraction. At the conclusion of the previous section (6.2), we had expressed in the Apparent Place system at T. (a,~) At time T, an observer observes a celestial body relative to the Observed Place system. The latter is defined by an origin defined by the observers position, a primary pole (Z) parallel to the true instantaneous celestial pole, a primary axis (X) parallel to the true vernal equinox, a Y-axis that makes the system right-handed. Evidently, the Observed Place is simply a translated Apparent Place system. Our task now is to move from the Observed Place to the Apparent Place where our mathematical models for position determination are formulated. This requires corrections for (i) diurnal aberration, (ii) geocentric parallax, and (iii) astronomic refraction. Diurnal aberration is the displacement of the direction to a celestial body due to the rotation of the earth. The diurnal constant of aberration, k, is expressed as a function of the earth's rotation (00 ), the e geocentric latitude of an observer geocentric position vector (p). k = 0.320 p cos~ = 0.0213 (~), and the length of the observer's This yields [e.g. Mueller, 1969] p cos~ (6-22) The changes in a celestial objects coordinates are given by 6aD = kS 1 cosh sec6 , (6-23) ~6D = k" sinh sin6 l • (6-24) 135 The convention adopted here for signs is ~aD ~<5D = = a AP <5 AP a Op , . T <5 0P • T T T (6-25) (6-26) This same convention applies to the corrections for geocentric parallax and astronomic refraction that are treated next. Geocentric parallax is the difference between the observed topocentric direction and the required geocentric direction (Figure 6-8). The changes, expressed in terms of the celestial objects right ascension and declination are ~aG =- (6-27) n sinh cosecz cos~ sec<5' n (sin~ cosecz sec<5' - tan <5 , cotz). (6-28) It should be noted that for celestial bodies other than the sun, ~<5G ~aG' are so small as to be negligible. Astronomic refraction is the apparent displacement of a celestial object lying outside our atmosphere that results from light rays being bent in passing through the atmosphere. In general, the light rays bend downward, thus a celestial object appears at a higher altitude than is really true (Figure 6-9). The astronomic refraction angle is defined by ~z R = z - z' (6-29) in whichz' is the observed zenith distance, z is the corrected zenith distance. The effects on a,<5 are ~a R ~<5R ~zR =- ~z =- ~zR R sinh cosec z I (sin~ cosec Zl cos~ sec<5' sec<5 1 - tan<5 1 cot (6-30) Zl). (6-31) values are usually tabulated for some standard temperature and pressure (usually 760mm Hg, T = 10°C, relative humidity 60%) • Corrections to those values are obtained through determinations of temperature, pressure, and 136 STAR 1T: GEOCENTRIC PARALLAX Z· : MEASURED DIRECTION Z : GEOCENTRIC DIRECTION GEOCENTRE Figure 6-8 Geocentric Parallax 137 -- -- ....... ""'"- ...... "'" " " u 1/ \ DIRECTION \ \ \ \ \ , \ Figure 6-9 Astronomic Refraction .. to S to S 138 relative humidity at the time of observation (T)o Finally, we can write the expressions for the right-ascension and declination of a celestial body as a AoPo T 0AoPo T = ~AoPo T - (~aD + ~ap + ~~ = 0'A.P· - (~OD + ~Op + in which a' ~ T and 0' ~ equation (6-21). ~OR) (6-32) , (6-33) refer to the Apparent Place coordinates obtained from This process is summarized in Figure 6-10. In closing this section, the reader should note the following. It is common practice in lower-order astronomical position and azimuth determinations to correct the observed direction z' for both geocentric parallax and refraction rather than use (6-32) and (6-33) above. Thus, in our math models we use z ~zp, = z' - (~z R + ~z P (6-34) ) the correction to a zenith distance due ..to geocentric parallax, is tabulated (e.g. SALS). Note that the effects of diurnal aberration are neglected. 6.4 Polar Motion The final problem to be solved is the transformation of astronomically determined positional coordinates (~,A) from the Apparent Place celestial system to the Average Terrestrial coordinate system. This involves two steps: (i) transformation of a AP ' 0AP (Apparent Place) to Instantaneous Terrestrial and (ii) transformation of WIT' AIT (Instantaneous Terrestrial) to Average Terrestrial. In the first step, we are moving from a non-rotating system to a coordinate system that is rotating with the ~arth. The Instantaneous 139 APPARENT PLACE (GEOCENTRIC) I ~ =rep-(A~ AP 6 AP +Arx. +A<X) AGO R =61 -(~6 +A6 +AS ) AP " DIURNAL ABERRATION AQh,llSD ~ GEOCENTRIC PARALLAX A~,A6G ~ ASTRONOMIC REFRACTION A~,A6R ~ OBSERVED PLACE (TOPOCENTRIC) Figure 6-10 Observed to Apparent Place . G 0 R 140 Terrestrial (IT) system is defined by -a geocentric origin, -a primary pole (Z) that coincides with the instantaneous terrestrial pole, -a primary axis (X) that is the intersection of the Greenwich Mean Astronomic Meridian and Instantaneous equatorial planes, -a Y-axis that makes the system right-handed. Comparing the definitions of the A.P. and I.T. systems, we see that the only difference is in the location of the primary axis, and futhermore, that XIT is in motion. Recalling the definition of sidereal time, we see that the A.P. and I.T. systems are related through GAST (Figure 6-11), namely X Y z X = R3 (GAST) (6-35) Y Z IT AP in which cos~ X = Y Z APT coscx X coso sino; Y sino APT z cos<S coso. = coso sino. , (6.36) sino or, using a spherical approximation of the earth X Y z IT = cos~ cosh cos~ sinh sin~ I.T. The reader should note that this transformation is usually (6-35) carried out in an implicit fashion within the mathematical models for position and azimuth determination rather than explicitly as is given here (see, for example, Chapters 8 and 9). The results that one obtains are then Figure 6-11 Transformation of Apparent Place To Instantaneous Terrestrial System 142 ~,h,A, all expressed in an Instantaneous Terrestrial coordinate system. The final step is to refer these quantities to the Average Terrestrial coordinate system. The direction of the earth's instantaneous rotation axis is moving with respect to the earth's surface. The motion, called polar motion, is counter- c1ockWis-e~ 'quas±--frt'egi:J.lar, ha~".:m aml'litude ,of'about· 5m',and' a period approximately 430 m.s.d. The motion is expressed in te~ of of the position of the instantaneous rotation axis with respect to a reference point The point used is the mean terrestrial pole for fixed on the earth's crust. the interval 1900-1905, and is called the conventional International Origin (C.I.O.). The Average Terrestrial (AT) coordinate system is the one which we would like to refer astronomic positions (~,h). This system differs in definition from the I.T. system only in the definition of the primary pole the primary A.T. pole is the C.I.O. Referring to figure 6-12, we can easily see that x x = R_2 y z (-x) R p 1 (-y) y P z AT (6-37) LT. Using a spherical approximation for the earth which is adequate for this transformation, one obtains [Mueller, 1969] cos~ cosh cos~ sinh sin~ = R_{-x ) -~ P A.T. R (-y ) 1 P cos~ cosh cos~ sinh sin~ (6-38) I.T. 143 y _0-----.--------... CIO y _ _- - - - - - - - -.. CIO I . II xp I I I I ~ ,/' ... -----------Yp INSTANTANEOUS TERRESTRIAL POLE x TRANSFORMATION FROM INSTANTANEOUS ·TO AVERAGE TERRESTRIAL SYSTEM. FIGURE 6-12 144 Specifically, one obtains after some manipulations, the two equations [Mueller, 1969] ~~ = ~AT - ~IT = yp ~A = AAT - AIT = -(xp (6-39) sinAIT - xp cosA IT sinAIT + yp cosAIT ) tan~IT (6-40) The effect of polar motion on the astronomic azimuth is expressed as [Mueller, 1969] (6-41) The complete process of position updating, and the relationship with all celestial coordinate systems is depicted in Figure 6-13. The relationships amongst all of the coordinate systems used in geodesy, with celestial systems in perspective, is given in Figure 6-14. Ecliptic Hour'Angle Right Ascension Horizon " Mean Celestial" @To Mean Celestial @T . True Celestial @T CELESTIAL COORDINATE SYSTEMS AND POSITION UPDATING [Krakiwsky and Wells, 1971] FIGURE 6-13 Apparent @T Observed @T I-' -I::'- VI TERRESTRIAL SYSTEMS CELESTIAL SYSTEMS ...... .~ 0'\ DIURNAL ABERRATION GEOCENTRIC PARALLAX REFRACTION - EULER ANGLES CU, i, Sl, i ORBITAL" SYSTEI\~, HELIOCENTRIC GEOCENTRIC I ITOPOCENTRIC COORDINATE SYSTEMS USED IN GEODESY FiBure 6-1L.. 7. DETERMINATION OF ASTRONOMIC AZIMUTH In Chapter 1, the astronomic azimuth was defined as the angle between the astronomic meridian plane of a point i and the astronomic normal plane of i through another point j. In Chapter 2 (Section 2.2.1), this angle was defined with respect to the Horizon celestial coordinate system in which the point j was a celestial object. Obviously, to determine the astronomic azimuth of a line ij on the earth, we must (i) take sufficient observations to determine the azimuth to a celestial body at an instant of time T, and (ii) measure the horizontal angle between the star and the terrestrial reference object (R.O.). In these notes, two approaches.to astronomic azimuth determination are studied: (i) the hour angle method, in which observations of Polaris (a Ursae Minoris) at any hour angle is treated as a special case, and (ii) the altitude method, in which observation of the sun is treated as a special case. Several alternative azimuth determination methods are treated extensively in Mueller [1969] and Robbins [1976]. The instruments·used for azimuth determination are a geodetic theodolite, a chronometer (e.g. stop watch), a HF radio receiver, a thermometer and a barometer. To obtain optimum accuracy in horizontal direction measurements, a striding level should be used to measure the inclination of the horizontal axis. Finally, the reader should note that the astronomic azimuth determination procedures described here yield azimuths that are classified as being second or lower order. This means that the internal standard deviation of several determinations of the astronomic azimuth would be, at best, O~5 to 1~5 [e.g. Mueller, 1969]. 147 Most often, using a 1" theodolite 148 with a striding 1eve1attachment, the standard deviation of the astronomic azimuth of.aterrestrial line would be of the order of 5" to 10" [e.g. Robbins, 1976]. 7.1 Azimuth by Star Hour Angles From the transformation of Hour Angle celestial coordinates (h,o) to Horizon celestial coordinates (A,a), we have the equation (2-20) tan A sinh. = sin~ (7-1) cosh - tano cos~ To solve this equation for A, we must know the latitude observation. ~ of the place of The declination of the observed star, 0, can be obtained from a star catalogue, ephemeris, or almanac (e.g. FK4, ~Mr SALS) and updated to the epoch of observation T. The hour angle of a star, h, can not be observed, but it can be determined if time is observed at the instant the star crosses the vertical wire of the observer's telescope. observes zone time (ZT), then the hour angle Assuming one is given by (combining equations (2-30), (3-11), (3-3), (3-13), (3-18» h = ZT + AZ + (a m h - 12 ) + Eq.E. + A - a, (7-2) in which ZT is observed, AZ and A are assumed known, a,nd are cata10guedin AA. . (or .0., .Eq. E and (~~ .-12 h ) m \(~m~12h:+_Eq·.E)iS ca.ta1ogued in SALS), and must be updated to the epoch of observation T. The minimization of the effects of systematic for any azimuth determination. errors are important This can be done, in part, through the selection of certain stars for an observing program. Assuming that the sources of systematic errors in (7-1) occur in the knowledge of latitude and the determination of the hour angle, we can proceed as follows. 149 Differentiation of (7-1) yields. dA = sinA cotz d~ + cos~ (tan~ (7-3) - cosA cotz)dh. Examination of (7-3) yields the following: when A (i) when (ii) = 0 ° or tan~ 180°, the effects of = cosA d~ are~9liminated, cotz, the effects of dh are "eliminated. Thus, if a star is observed at transit (culmination), the effects of minimized, while if a star d~ are is observed at elongation (parallactic angle ° p = 90 ), the effects of dh are minimized. Of course, it is not possible to satisfy both conditions simultaneously; however, the effects will be elliminated if two stars are observed such that sinAl cotz1" =-sinA2 (7-4) cotz 2 - (COSAl cotz l + COSA2 cotz 2 ) 2tan~ =0 (7-5) As a general rule, then, the determination of astronomic azimuth using the hour angle method must use a series of star pairs that fulfill conditions (7-4) and (7-5). This procedure is discussed in detail in, for example, Mueller [1969] and Robbins [1976]. A special case can be easily made for cicumpolar stars, the most well-known of which in the northern hemisphere is Polaris (a. Ursae'Minoris ). When ~ > 15° Polaris is easily visible and directions to it are not affected unduly by atmospheric refraction. eliminated. cos~ = 0°, the error d~ is Furthermore, since in (7-3), the term [Robbins, 1976] (tan~ then when ~ Since A - cosA coti) = 90° I = cos~ cosz cosp, the effects of dh are elliminated. (7-6) Then Polaris can be observed at any hour angle for the determination of astronomic azimuth by the hour angle method. 150 A suggested observing sequence for Polaris is as follows [Mueller, 1969]: (i) (ii) (iii) (iv) (v) Direct on R.O., record H.C.R., Direct on Polaris, record H.C.R. and T, Repeat (ii), Repeat (i)~ , Reverse telescope and repeat (i) through (iv). The above observing sequence constitutes one azimuth determination; eight sets are suggested. Note that if a striding level is 'used, readings of both ends of the level (e.g. e and w for direct, e " and Wi should be made and recorded after the paintings on the star. for reverse) An azimuth correction, for each observed set,is then given by [Mueller, 1969] I:J.A" = d" ( (w + 4 Wi) - (e + e ' ) ) cotz (7-7) in which d is the value in arc-seconds of each division of the striding level. Briefly, the computation of astronomic azimuth proceeds as follows: (i) for each of the mean direct and reverse zone time (ZT) readings of each set on Polaris, compute the hour angle (h) using (7-2), (ii) using (7-1), compute the astronomic azimuth A of Polaris for each of the mean direct and reverse readings of each set, (iii) using the mean direct and mean reverse H.C.R.'s of each set on Polaris and the R.O., compute the astronomic azimuth of the terrestrial line, (iv) the mean of all computed azimuths (two for each observing sequence, eight sets of observations) is the azimuth of the terrestrial line, 151 (v) computation.of the standard deviation of a single azimuth determination and of the mean azimuth completes the computations. computations may be shortened somewhat by (i) using the mean readings direct and reverse for each set of observed times and H.C.R.'s (e.g. only 8 azimuth determinations) or (ii) using the mean of all time and H.C.R.'S, make one azimuth determination. If either of these approaches are used, an azimuth correction due to the non-linearity of the star's path is required. This correction (6A ) is termed a second order c curvature correction [Mueller, 1969]. A" c It is given by the expression n = I m. (7-8) ~ i=l in which n is the number of observations that have been meaned (e.g. 2 for each observing sequence (direct and reverse), 16 for the total set of The term CA is given by observations (8 direct, 8 reverse». = 2 2 cos h - cos, A tanA (7-9) 2 . 2h cos A s~n in which the azimuth A is the azimuth of the star (Polaris) computed without the correction. The term m. is given as ~ 2 mi = ~"[i "[i = sin 2" (7-10) , where (T. - T ) ~ 0 and T 0 = Tl + T2 + n , (7-11) .... + Tn (7-12) If, when observing Polaris, the direct and reverse readings are made within 2m to 3m , the curvature correction ~A c will be negligible and 152 Instruments: K ern DKM3-A theodolite (No. 82514); Hamiltonsidereal chronometer (No. 2EI2304); Favag chronograph with manual key; Zenith transoceanic radio. Azimuth Mark: West Stadium Observer: A. Gonzalez-Fletcher Computer: A. Gonzalez-Fletcher Local Date: 3-31-1965 Location: OSU old astro pillar tit e! 40°00'00" -51132~0· Ae! Star Observed: FK4 No. 907 (a Ursae Minoris [Polaris]) !"\ 1. Level Corrections (Sample) AA , J cotz (e+e) ' = 4'd [ ,(w+w) Determination No. 2 3 0'!347 41.7 43.3 -0'!59 0'!347 41.9 42.8 -0'!31 1 d/4 cot z w+vI e+ff AA d 0'!347 41.8 43.3 -0'!52 = 1 '!6/division 2. Azimuth Computation (Sample) tan A == sin h/ (sin If, cos h - cOs c;l) tan 6) " 0' .. Determination No. I 1 j\Iean chronometer reading ~f direct and reverse point1O"'S 12 Chronometer correction (computed similar to Example 8.4) 3 AST 4 a(8.J?parentiJ .' ,,',~ 5 h == AST - a 6 h (arc) 7 6 (apparent 9 tan 2 91:22:02~35 9~6t;,13~85 101111 "38~ 16 -5c51~25 -5"51!28 9"s0"22~ 57 10b05"46~83 9hI6:11~ 10 3 -5rsl~33 Ih57"53~46 Ih57"53~46 ih57rs3~46 7h18c17~64 7~2"29~11 8"o7=S3~37 109~4'24':60 113°07'16'!65 121°58'20'!55 1. 8 cos h 10 11 12 13 14 15 1 6 sin'cp cos h cos 9 tan 6 (10) - (11) Sill h tan A == (13)/(12) A (at the average h) 89°06'12'!92 -0.33501582 63.91162817 -0.21534402 48.95914742 -49.17449144 0.94221251 -0.01916059 35S054'08'!33 89°06'12':92 -0.39267890 63.91162817 -0.25240913 48-:95914742 -49.21155655 0.91967535 -0.01868820 35S055 '45'!73 89 006'12'!92 -0.529510321 63.91162812 -0.3403626; 48.95914742 -49.299510091' 0.84830350 -0.01720714 i 359°00'51'!12! AZIMUTH BY THE HOUR ANGLE OF POLARIS [Mueller, 1969] FIGURE 7-1 I 153 16 Time difference between direct and reversed ~)ointings (21) 17 Curvature correction (equation (9;17» 18 A (polaris) (15) + (17) 19 Circle (hor~zontal) reading on Polaris 20 Correction for dislevelment (AA) 21 Corrected circle reading (19) + (20) 22 Circle reading on Mark 23 Angle between Mark and Polaris (22) - (21) 24 Azimuth of Mark = (18) + (23) .. I = = = 7"12.'4 2D44~5 2~6~5 0'15 35So54 'OS ':S O'!1 35S055 '45':S 0':1 359000'51'!2 25So..zS '48':9 258°27'28':2 258°32'39'!0 ! -O'!5 -0':6 -0':3 25S~5'4S'!4 00~1'13':7 25S~7 '27':6 00°01 '14'!7 258°32'3S'!7 00001'17'!5 101°35'25'!3 101~3'47'!1 101°28 '3S '!8 . 100~9'34'!1 100~9'32'!9 100~9'30'!0 i 3. Final Observed Azimuths from Eight Determinations Determination No. Azimuth of Marlt 1 2 -. 3 4 5 6 7 8 Final (mean) 100°29 '34 '!1 32.9 30.0 31.3 32.0 ." 30.3 31.6 31.5 100°29'31':71 v -2'.'39 -1.19 1.71 0.41 -0.29 1.41 0.11 0.21 [v]=-O'!02 Standard deviation of an azimuth det.ermination: mA =jrvv] n-1 = J12.35 8-1 = 1"33 . Standard deviation of the mean azimuth: M m. ..--.... A=rn 1':33 =18 = 0'.'47 Result: FIGURE 7-1(continued) vv 5.71 1.42 2.92 0.17 O.OS 1.99 0.01 0.04 [vv]=12.35 154 could therefore be neglected [Robbins, 1976]. An example of azimuth determination by the hour angle of Polaris is given in Figure 7-1. Azimuth by hour angles is used for all orders of astronomic work. The main advantages of this method are that the observer has only to observe the star as it coincides with the vertical wire of the telescope and since no zenith distance measurement is made, astronomic refraction has no effect. The main disadvantages are the need for a precise time-keeping device and a good knowledge of the observer's longitude. Table 7-1 summerizes, for several situations, the sources of errors and their magnitudes in the determination of astronomic azimuth by stars hour angles. 7.2 Azimuth by Star Altitudes Azimuth by star altitudes yields less accurate results than azimuth by star hour angles. The two reasons for this are (i) the star must be observed as it coincides with both the horizontal and vertical wires of the telescope, and (ii) the altitude observation is subject to the effects of astronomic refraction. The method does have the advantage, however, that a precise knowledge of the observer's longitude is not required and an accurate time-keeping device is not needed. Azimuths determined by star altitudes are not adequate for work that requires 0A to be 5" or less. 155 Stars at Elongation .. .. ·. ·. ·. .. .. ·. ·. ·. a 15 35° Polaris 4» Prime Vertical 15° Lower Transit 22°.5 15° 60° 75° 03 sec a (vertical wire on star) 2".4 2".3 2".6 2".7 2".6 04 (m=asurement of horizontal angle) 2".5 2".5 2".5 2".5 2".5 05 tan a (reading of plate level) 3".5 2".9 1".3 2".1 1".3 0".0 0" .1 1".5 111.5 0".8 4".9 4".5 4".1 4".5 3".9 0".9 0".7 0".3 0".5 0".3 3".6 3".5 3',' .9 4".0 3".7 aT (time) ~um 06 (plate ·. ·. ·. level) •• ·. tan a (reading of striding level) Sum ( striding level) ·. Azimuth by hour angle: random errors in latitude 30° Stars at a 15 .. .. ·. .. . . ·. Elongation 64° Polaris 4» Prime Vertical 22°.5 75° Lower Transit 20° 30° 40° 50° 60° 70° 03 sec a(vertical wire on star) 4".6 4".0 2".7 2".7 2".9 , 3".3 04 (measurement of horizontal angle) 2".5 2".5 2".5 2".5 2".5 2".5 tan a(reading of plate level) 10".2 8".7 211.1 1".8 211.9 4".2 0".0 0" .1 2".6 2".1 1".7 1" .3 11".5 9".9 5".0 4".6 5" .1 6" .0 2".6 2".2 0".5 0".5 0".7 1".0 5".9 5".2 4".5 3".3 4".2 4".5 05 aT (time . .. Sum (Plate level) 06 tan a (reading of striding level Sum (Striding level) Azimuth by hour angle: random errors in latitude 60° Table 7-1 [Robbins, 1976] 156 From a relationship between the Horizon and Hour Angle coordinate systems (egn. (2-26), rearranged) = cosA sino - cosz sin4'> (7-13) sinz cos4'> or with z = 90 - a, cosA = sin6 - sina sin4'> (7-14) cosa cos4'> To solve either (7-13) or (7-14) for A, the latitude 4'> of the observer must be known. The declination, 6, of the observed celestial body is obtained from a catalogue, ephemeris, or almanac and updated to the time of observation. The zenith distance (or altitude) is the measured quantity. The main sources of systematic 'error with this method include error in latitude (assumed) and error in the reduced altitude due to uncorrected refraction. dAsinA = Differentiating (7-14) yields (tana - cosA tan4'» + d4'> (tan4'> - cosA tana) da (7-15) The effects of d4'> are zero when tana • wh1ch occurs when h = cosA tan4'> h = 90 0 0 (7-16) h (6) or 270(18). The effects of da are zero when tan4'> = cosA tana (7-17) which occurs when the observed celestial body is at elongation. To minimise the effects of systematic errors, two stars are observed such that al = a2 Al = 360 0 (7-18) ' - A2 (7-19) 157 Further details on the selection of star pairs observing procedure and the computation of azimuths, can be found in, for example, Mueller [1969] and Robbins [1976]. using Polaris, observations should be made at elongation. Note that no star should have an altitude of less th~ 15° when using this method, thus for Polaris, ~>lso. Estimates of achievable accuracy are given in Table 7-2. The method of azimuth determination by altitudes is often used in conjunction with sun Observations. Using a I" theodolite and the observation and computation procedures described here, order of 20". 0A will be of the When special solar attachments are used on an instrument, such as a Roelofs solar prism [Roelofs, 1950], and accuracy (oA) of 5" may be attained. A complete azimuth determination via sun observations consists of two morning and two afternoon determinations. It is best if 30° ~ a ~ 40°, and it should never be outside the range 20° ~ a < 50°. Figure 7-1 illustrates, approximately, the optimum observing times. A suggested observing procedure for one azimuth determination is as follows [Robbins, 1976): (i) Direct on RO, record H.C.R. and level (plate or striding) readings, (ii) Direct on Sun, record H.C.R. and V.C.R., and time to nearest m 1 (for methods of observing the sun, see, for example, Roelofs (1950), record level readings, (iii) Reverse on Sun, record H.C.R. and V.C.R., and time to nearest m 1 , record level readings, 158 h = 90° Prime vertical Stars at Elongation · . ·. ·. ·. ·. ·. ·. 40° 25° 35° 51° 58° 17° 0'1 (pointing of horizontal wire on star) 0" 1".0 1".7 0'2 (measuring of altitude) 0" 1".3 2".3 O'R (determination of refraction) 0" 0".7 1".2 0'3 sec a (pointing of vertical wire) 2".6 3".3 3".7 0'4 (horizontal angle measurement) 2".5 2".5 2".5 0'5 tan a (reading plate level) 4".2 2".3 3".5 5" • 5 5".1 6".5 0'6 tan a (reading striding level) 1".0 0".6 0".9 pum (Striding level) 3".7 4".5 5".5 let 0 ·. ~um (Plate level) ·. Azimuth by altitude: random errors in latitude 30° Stars at a 15 Elongation ·. ·. ·. ·. · . ·. ·. ·. 0'1 (pointing of horizontal wire on star) h = 90° Prime vertical 64° 35° 35° 75° 41° 30° 0" 0'2 (measuring of altitude) -+ -+ O'R (determination of refraction) -+ 0'3 sec a (pointing of vertical wire) -+ 0'4 (horizontal angle measurement) -+ 2".5 -+ 10".3 -+ 2".5 -+ 3".5 + 11".6 + 10".2 -+ 0'6 tan a (reading striding level) + 2".6 -+ 0".9 -+ 0".9 Sum (striding level) + 5".8 + 9".6 -+ 0'5 tan a (reading plate level) Sum (plate level) ·. 0" -+ 4".8 5".2 -++ 6".8 -+ 6".2 - 0" -+ 3".2 -+ 3".5 4".6 -+ 3".7 -+ 3".7 Azimuth by altitude: random errors in latitude 60° Table 7-2 [Robbins, 1976] + 2".5 -+ 3".5 10".8 10".3 159 __Yv_~ .".. ."..-"¥ . . . . ...... , SUN MOVING TOO FAST APPROX. ,,0 AM-2 PM ..... " " ,, , ~" \ I \ I REFRACTION , TOO GREAT " , REFRACTION TOO GREAT Figure 7-2 Azi~uth by the Sun's Altitude 160 (iv) (v) Reverse on R.O., record H.C.R., record level readings, Record temperature and pressure. The computation procedure resulting from the above observations is: (i) Mean Direct and Reverse horizontal readings to the R.O. and to the Sun, (ii) correct the horizontal directions in (i) for horizontal axis dislevelment using equation (7-7), (iii) (iv) compute the angle Sun to R.O., mean direct and reverse zenith distance (or altitude) measurements to the Sun, then correct the mean for refraction and parallax, (v) determine the time of observation in the time system required to obtain a tabulated value of 15 of the sun, (vi) compute the updated value of 0 for the time of observation (tabulated 0 plus some correction for time), (vii) (viii) using either (7-13) or (7-14), compute the azimuth to the sun, using the angle determined in (iii), co~pute the azimuth to the R.O. The following example illustrates the observing and computation procedures for an azimuth determination via the altitude of the sun. 161 Ex~~ple: Azimuth via Sun's Altitude (using SALS). Instrument: A Date: May 6, 1977 B Time: Central Daylight Time (90 0 W) R.O.: Obs. Procedure: Inst. Latitude (~): 38° 10' 10" Sun observed in quadrants. V.C.R. H.C.R. Sight Time Remarks B 0° 00' 02~0 Direct Sun 157° 54' 56~0 56° 12' 00" ISh 40m Temp. 20°C Reverse Sun 339° 05' 30~0 57° 10' 00" ISh 46m Pres. 1000 mh. B 180° DO' 04~0 Note: No level readings required. Means: H.C.R. Sun: 30' l3~0 B: DO' 03~0 30' 10~0 H.C.R. Horiz. Angle: 158 ° V.C.R. 56° 41' . 00" .Sketch: 8 (R.O.) Reduction of Altitude Mean Altitude (a) = S6° 90° l-1ean Refraction (r ) ° Correcting factor ( f) 41' 00" = 33° 19' ~O'' l' 24" = 88" = 0.95 Refraction == r xf == 88" xO.95 ° Parallax = + Corrected Altitude = 33° == 06" 17' 42" 162 computation of Sun's Declination 15h Observed (Mean) Time 43m _Olm Clock Correction III OOm + 6h OOm 20h 42 m 16° 39~0 Correction for Daylight Time Time Zone U.T. Sun's 0 at 6 May 18h U.T. Change in 0 since 18h (N) U.T. (2 h 42 m) Sun's 0 at 6 May 20h 1!9 + 16° U.T. 40!9 Azimuth Computation cosA = sin 0 - sin a sin cos a cos~ cosA = sin(16° ~ 40!9) - sin (33 0 cos (380 (7-14) 17' 17' 42") sin (38 0 42") cos (38 0 10' 10' 10") 10") - 0.52191 = - 0.0794215 0.657391 + Azimuth of Sun =cos -1A = 360°- 94° 33' 19" = 265 0 26' 41" Mean Horiz. Angle = 158 0 30' 10" Azimuth to B (R.O.) = 106° 56' 31" = 8. DETERMINATION OF ASTRONOMIC LATITUDE Astronomic latitude was defined in Chapter 1. To deduce procedures for determining astronomic latitude from star observations, we must examine an expression that relates observable, tabulated, and known quantities. Equation (2-22), involving the transformation of Hour Angle coordinates to Horizon coordinates, namely cosz = sin~ sino + cos~ coso cosh , requires that zenith distance (z) and time (h) be observed, that declination (0) and right ascension (a) be tabulated, and that longitude solve for an unknown latitude (~). (A) be known to The effects of systematic errors in zenith distance and time, dz and dh'respectively, are shown via the total derivative of (2-22), namely d~ When A (secA = 0, = -1, then d~ = = -dz tanA =0). - secAdz (secA cos~ = 1, tanA tanA dh = 0), and when A (8-1) = 180o , then d~ = dz For this reason, most latitude determinations are based on zenith distance measurements of pairs of stars (one north and one south of the zenith such that zn = -z~) s as they transit the observer's meridian. The instrumen~used for latitude determination are the same as those for azimuth determination: a geodetic theodolite with a striding level attachment, a chronometer, an HF radio receiver, a thermometer and a barometer. Two astronomic latitude determination procedures are given in these notes: (i) Latitude by Meridian Zenith Distances, (ii) Latitude by Polaris at any Hour Angle. The accuracy (o~) of a latitude determination by either method, using the procedures outlined, is 2" or better [Robbins, 1976]. For alternative latitude determination procedures, the reader is referred to, for 163 164 example, Mueller [1969] and Robbins [1976]. 8.1 Latitude by Meridian Zenith Distances Equations (2-54)., (2-55), and (2-56), rewritten with subscripts to refer to north and south of zenith stars at transit are ~ = 5N - z ~ = z ~ = 180 0 - S N - 5s (UC north of zenith) , (8-2) (UC south of zenith) , (8-3) 5 - z (LC north of zenith). N N (8-4) If a star pair UC north of zenith - UC south of zenith are observed, then a combination of (8-2) and (8-3) yields (8-5) while a star pair LC north of zenith -UC south of zenith gives ~ =~ (5 S - 15 ) + ~ (z N S - z ) N + 90 (8-6) 0 Equations (8-5) and (8-6) are the mathematical models used for latitude determination via meridian zenith distances. An important aspect of this method of latitude determination is the apriori selection of star-pairs to be observed or the selection of a star programme. Several general points to note regarding a star programme are as follows: (i) more stars should be listed than are required to be observed (to allow for missed observations due to equipment problems, temporary cloud cover, etc.), (ii) the two stars in any pair should not di~fer in zenith. distance by more o than about 5 , (iii) the stars in any pair, and star pairs, should be selected at time intervals suitable to the capabilities of the observer. 165 For details on star programmes for latitude determination by meridian zenith distances, the reader is referred to, for example, Robbins [1976]. A suggested observing procedure is: (i) set the vertical wire of the instrument in the meridian (this may be done using terrestrial information e.g. the known azimuth of a line, or one may determine the meridian via an azimuth determination by Polaris at any Hour Angle), (ii) set the zenith distance for the first north star; when the star enters the field of view, track it with the horizontal wire until it reaches the vertical wire, (iii) (iv) record the V.C.R., temperature, and pressure, repeat (i) to (iii) for the south star of the pair. This constitutes one observation set for the determination of Six to eight sets are required to obtain (J 4> = 2" or less. ~. The computations are as follows: (i) (ii) correct each observed zenith distance for refraction, namely for each star pair, use either equation (8-5) or (8-6) and compute (iii) ~, compute the final ~ as the mean of the six to eight determinations. An example of this approach can be found, for instance, in Mueller [1969]. 8.2 Latitude by Polaris at any Hour Angle Since Polaris is very near the north celestial pole, the polar distance (P = 90-~, ° very close to O. Figure 8-1) is very small (P<lo), and the azimuth is Rewriting equation (2-22) (Hour Angle to Horizon coordinate transformation) with z sina = sin~ cosp + = (90 - a) and P cos~ sinp cosh = 90 - ~ yields (8-7) 166 Now, setting ~ = (a + ~a), in which oa and P are differentially small, and 'substituting in (8-7) gives Figure 8-1 Astronomic Triangle for Polaris sin a = sin (a + c5a) cosP + cos (a + oa) sinP cosh .' = sina cosoa cos.P + cosa sinc5a cos P + cosa cosoa sinP cosh - sina sin6a sinP cosh. (8-8) Then, replacing the small angular quantities (6 a and P) by their power . series (up .to and including 4th order terms), namely sine cose (8-8) =e _e 3 + =1 - ..... , -4:6 4 - 3: 62 2: + , becomes sina = . 3 (sina(l- + cosa (oa - + 2! + (cosa (1 4! oa p2 » (1- sina (oa - + 2! 3! ~~ 3 » p3 (P - 3: ) cosh. Now, taking only the first-order terms of (8-9) into account yields sina = sina + oa cosa + Pcosa cosh (8-10) .• (8-11) or c5a = -Pcosh Replacing the second-order c5a terms in (8-9) with (8-11) above, and neglecting all terms of higher order gives (8-9) 167 oa = - Pcosh + . 2h tana sJ.n (8-l2) Repeating the above process for third and fourth order terms yields the final equation ~ =a + oa =a - Pcosh + sinP tana sin2 h P 2 P . 3 . 4h 3 . 2P coshsJ.n ' 2h + 8 3P sJ.n sJ.n P sJ.n tan a (8-l3) Except in very high latitudes, the series (8-13) may be truncated at the third term. A suggested observation procedure is as follows: (i) observe polaris (horizontal wire) and record V.C.R., time, temperature and pressure; repeat this three times. (ii) reverse the telescope and repeat (i). This constitutes the observations for one latitude determination. It should be noted that if azimuth is to be determined simultaneously (using Polaris at any hour angle), then the appropriate H.C.R.ts must be recorded (such a programme requires that the observer place Polaris at the intersection of the horizontal and vertical wires of the telescope). The computation of latitude proceeds as follows: (i) (ii) compute mean zenith distance measurement for six measurements, correct mean zenith distance for refraction and compute the mean altitude (a = 90 - z), (iii) compute the mean time for a, then using (7-2), compute the mean h (note that mean a and (a m - 12h) must be computed for this part of the latitude determination), 168 (iv) compute a mean (v) compute ~, then P = 90 - 0 Ip A mean of 12 to 20 determinations will yield an astronomic latitude with = 2" alp or less [Robbins, 1976]. The reader should note that the use of a special SALS table (Pole Star Tables) leads to a simple computation of latitude using this method, namely (8-14) in which a is the corrected, meaned altitude, and a o ' aI' a 2 are tabulated values. (8-13) Using this approach results will not normally differ from those using by more than O,:~'2 (12"), while in most cases the difference is within O!l (6") [Robbins, 1976]. Both computation procedures are illustrated in the following example taken from Robbins [1976]. Example: (i) Latitude from Observations of Polaris at any Hour Angle with SALS tables Date: 23 August 1969 cj>=57003' A.= 7° 27' W =- Oh 29m 48 s Mean of Observations: a = 56° 57' 24" Press. = 1009mb. Computations: h RCa -12 +Eq.E) m = 169 GAST :::::: UT + R + R "" 20h 09m 45~4 oh 29m 48 5 19h 39m 57~4 A LAST a :::: 56° 57' 24" (mean refraction) r :::: 38 " ° f a ~ (refraction factor) :::::: corr == a corr a - (r + a 0 0 1.00 x f) == 56° 56' 46" + a l + a 2 == 56° 56 1 46" ~ (if) ::::: :::: 57 ° 02' with equation (8-13) + 5'48" 43" (first three terms only) as in (i) plus: (l == 2h 03 m 09 5 17" a. a ::::: 56° p 3163" 4? ::::: _. 56' 56° 56' 46 11 46" + + 09" 319~5 ~ + = 57° 36~9 02' 42~'4 + 00" 9. DETERMINATION OF ASTRONOMIC LONGITUDE In chapter 3, we saw that l~ngitude is related directly with sidereal time, namely (egn. 3-3 ) (9-l) A = LAST - GAST From the Hour Angle - Right Ascension systems coordinate transformations (egn. 2-31) LAST =a (9-2) + h • Replacing LAST in (9-l) with (9-2) yields the expression A = a + h - GAST (9-3) This equation (9-3) is the basic relationship used to determine astronomic longitude via star observations. The right ascension (a) is catalogued, and the hour angle (h) and GAST are determined respectively via direction and time observations. Let us first examine the determination of GAST. Setting TM as the observed chronometer time (say UTC), and 6T as the total chronometer correction, then GAST = TM + 6T (9-4) The total correction (6T) consists of the following: (i) the epoch difference 6T o at the time of synchronization of UTC and TM times, (ii) (iii) (iv) chronometer drift 6 1 T, DUTl, given by DUTl = UTl - UTC (see Chapter 4), the difference between UTl and GAST at the epoch of observation given by h LMST = MT LAST = LMST + Eq.E. + (aM - 12 ) Finally, equation (9-3) reads A =a + h - (TM + 6T) 170 (9-5) 171 The determination of the hour angle (h) is dependent on the astronomic observations made. coordinate transformation From the Hour Angle - Horizon systems (eqn. 2-22), = cosh cosz - sino coso sin~ (9-6) cos~ can be used for the determination of a star's hour angle. and TM as the observed quantities, ~ Now, taking z as a known quantity, and 0 as catalogued, the sources of systematic errors affecting longitude determination will be in zenith distance (dz), the assumed latitude time (TM) and the chronometer correction (d6T). (d~), the observed Replacing h in (9-5) by (9-6), and taking the total derivative yields dA =- (sec~ cotA d~ + sec~ cosecAdz + dT + d6T) • (9-7) Examining (9-7), it is obvious that the errors dT and d6T contribute directly to dA and can not be elliminated by virtue of some star selections. other hand, (i) and dz can be treated as follows: when the observed star is on the prime vertical (A (ii) d~ On the = 90 o or 270 0 ) , then d~ =0 the effects of dz can be elliminated by observing pairs of stars such that zl=z2 • As with azimuth and latitude determination procedures in which pairs of stars have to be observed under certain conditions, a star program (observing list) must be compiled prior to making observations. For details, the reader is referred, for example, to Mueller [1969] and Robbins [1976]. The primary equipment requirements for second-order (0). = 3" or less) longitude determination are a geodetic theodolite, a good mechanical or quartz chronometer, an HF radio receiver, and a chronograph equipped with a hand tappet. 172 Finally, before describing a longitude determination procedure, the reader should note the following. Since the GAST used is free from polar motion, LAST must be freed from polar motion effects. by adding a polar motion correction (6A p This is accomplished : see Chapter 6) to the final determined longitude (not each observation), such that the final form of equation (9-5) becomes A 9.1 =a (9-8) + h + 6Ap - (TM + 6T) Longitude by Meridian Transit Times When a star transits the meridian, h If pairs of stars are observed such that zl = (9-7) are e11iminated. = oh d~ = LAST. z2' then the effects of dz in The observation of time (TM) as a star transits the meridian enables longitude computation using (9-8). error will be (or l2h) and a The main sources of and the observation of the stars transit of the meridian (due partly to inaccurate meridian setting and partly to col1iIiiation). The " latter errors cause . :. a timing error. A correction for this effect is computed from [Mueller, 1969] dA db or = db= cos 15 cosec z (9-9) sin z cos '8 -dA (9-10) where dA is the rate for the star to travel I' (4 s ) of arc in azimuth, yielding db in seconds. Example For dA = I' = 4s dbN s = 5.85 db s = 0~80. Then, if the longitude difference determined from north and south s 2.0, the corrections to A are as follows: stars is 173 (2.0 x 5.85) 5.85 + 0.80 dlLA s = (2.0 x 0.80) = = --.!.=-:..::.....;:;;:.....;:;..;;....::.=..!- 0 s• 2 4 (5.85 + 0.80) A suggested field observation procedure for the determination of longitude by meridian transit times is [Mueller, 1969]: (i) make radio-chronometer comparisons before, during, and after each set of observations on a star pair, (ii) set the instrument (vertical wire) in the meridian for the north star and set the zenith distance for the star; when the star enters the field of view, track it until it coincides with the vertical wire and record the time (hand tappet is pressed), (iii) set the instrument for the south reverse the (iv) star of the pair (do not and repeat (ii), instrumen~ repeat (ii) and (iii) for 12-16 pairs of stars. The associated computation procedure is as follows: (i) (ii) (iii) (iv) compute the corrected time for each observed transit (e.g. TM + compute the apparent right ascension for each transit, compute the longitude for each meridian transit using (9-8), compute dh for each star; compute .hA ~ AN-As compute the correction to longitude (dIl) for each star pair; for each star of a pair (see example), (v) (vi) compute the mean longitude from the 12-16 pairs, apply the correction hA p An 11.2] • ~T) to get the final value of longitude. example of this procedure is given in Mueller [1969; Example , 174 In closing, the reader should be aware that there are several other methods for second order longitude determination. For complete coverage of this topic, the reader is referred to Mueller [1969) and Robbins (1976). 175 REFERENCES Astronomisches Rechen-Institut, Heidelbe,rg [1979]. A,Eparent Places of Fundamental~ta~s',l,981 : Containing 'the-:1535 "Sta~$':"irt;,tbe: Fourth FtiIidame,tibil ;Catalogue~', (APFS) ~ 'Ver1agG~ Braun; ;Kar1sruhe. Boss, B. [1937]. General Catalogue of 33342 stars for the epoch 1950.0, (GC). publications of the Carnegie Institute of Washington, 468. Reissued by Johnson Reprint Co., New York. Eichorn, 'H. [1974]. Astronomy of Star Positions. Co., New York. Frederick Ungar Publishing Fricke, W. and A. Kopff [1963]. Fourth Fundamental catalogue resulting from the revision of the Third Fundamental Catalogue (FK3) carried out under the supervision of W. Fricke and A. Kopff in collaboration with W. Gliese, F. Gondolatsch, T. Lederke, H. 'Nowacki, W. Strobel and P. Stumpff, (FK4). Veroeffentlichungen des Astronomischen ReckenInstituts, Heidelberg, 10. H.M. Nautical Almanac Office [lg8~]. The star Almanac for Land Surveyors for the year 1981. Her Majesty's Stationery Office, London. (in Canada: Pendragon House Ltd. 2525 Dunwin Drive, Mississauga, Ontario', L5L 1T2.) Heiskanen, W.A. and H. Moritz [1967]. Company, San Francisco. Physical Geodesy. W.H. Freeman and Krakiwsky, E.J. and D.E. Wells [1971]. Coordinate Systems in Geodesy. Department of Surveying Engineering Lecture Notes No. 16, University of New Brunswick, Fredericton. Morgan, H.R. [1952]. Catalog of 5268 Standard Stars, 1950.0. Based on the Norman System N30. Astronomical Papers Prepared for the Use of the American Ephemeris and Nautical Almanac, 13, part 3. Government Printing Office, Washington, D.C. Morris, William (editor) [1975]. The Heritage Illustrated Dictionary of the English Language. American Heritage Publishing Co., Inc. Mueller, Ivan I. [1969]. Spherical and Practical Astronomy as Applied to Geodesy. Frederick Ungar Publishing Co., New York. Robbins, A.R. [1976]. Military Engineering, Volume XIII-Part IX, Field and Geodetic Astronomy. School of Military Survey, Hermitage, Newbury, Berkshire, U.K. Roelofs, R._[1950]. Astronomy Applied to Land Surveying. Ahrend & Zoon, Amsterdam. N.V. Wed. J. U.S. Naval Observatory, Nautical Almanac Office [19'801. The Astronomical Almanac fo"'( the Year 19.81; U.S'.' Governm~nt P"'(intina Offi~, Washington, , D.C. 176 APPENDIX A REVIEW OF SPHERICAL TRIGONOMETRY In this Appendix the various relationships between the six elements of a spherical triangle are derived, using the simple and compact approach of A.R. Clarke, as given in Todhunter and Leathem ("Spherical Trigonometrytt, Macmillan, 1943). The 27 relations derived are listed at the end of the Appendix • A.1 Derivation of Relationships In Figure A-l, the centre of the unit sphere, 0, is joined to the vertices A, B, C of the spherical triangle. Q and R are the projections of C onto OA and OB, hence OQC and ORC are right angles. P is the projection of C onto the plane AOB, hence CP makes a right angle with every line meeting it in that plane. OPC are right angles. Thus QPC, RPC, and Note that the angles CQP and CRP are equal to the angles A and B of the spherical triangle. We show that OQP is also a right angle by using right triangles COP, COQ, CQP to obtain C02 = Op2 + PC 2 (A-1) C02 = OQ2 + QC 2 (A-2) QC 2 = Qp2 + PC2 (A-3) Equating (A-l) and (A-2), and substituting for QC 2 from (A-3) Op2 = OQ2 + Qp2 that is OQP is a right angle. right angle. Similarly it can be shown ORP is also a 177 c A _ _ _ _ _----8 Figure A-1 178 In Figure A-2, on the plane AOB, S is the projection of Q on OB and T is the projection of R on OA. Note that the angle AOB is equal to the side c of the spherical triangle, and that angles AOB SQP = TRP. PC = RC = Then from Figure A-1 sin B = QC sin A (A-4) and from Figure A-2 OR = OS QP sin c (A-5) RP = SQ - QP cos c (A-6) QP = TR - RP cos c (A-7) + We can now make the substitutions OR = cos a RC = sin a OQ = cos b QC QP RP = sin b = QC cos = RC cos A = sin b cos A B = sin a cos B OS = OQ cos c = cos b cos c SQ = OQ sin c = cos b sin c OT = OR = OR cos c = cos = cos a cos c TR sin c a sin c to obtain from (A-4) to (A-7) sin a sin B cos a = cos = sin b sin A (A-8) b cos c + sin b sin c cos A (A-9) sin a cos B = cos b sin c - sin b cos c cos A sin b cos A = cos a sin c - sin a cos c cos B Equation (A-8) is one of the three Laws of Sines. one of the three Laws of Cosines for Sides. (A-10) (A-11 ) Equation (A-9) is Equations (A-lO) and 179 o 8 c Figure A-2 180 (A-11) are two of the twelve Five-Element Formulae. The Four-Element Formulae can be derived by multiplying Equation (A-10) by sin A, and dividing it by Equation (A-8) to obtain sin A cot B = sin c cot b - cos A cos c rearranged as cos A cos c = sin c cot b - sin A cot B (A-12) Similarly multiplying Equation (A-11) by sin B and dividing by the transposed Equation (A-B) we obtain cos B cos c = sin c cot a - sin B cot A (A-13) Equations (A-12) and (A-13) are two of the six Four-Element Formulae. It can be shown that all the above Laws remain true when the angles are changed into the supplements of the corresponding sides and the sides into the supplements of the corresponding angles. Applying this to Equations (A-8) , (A-12), and (A-13) will not generate new equations. cos a However, when applied to Equation (A-9) = cos b cos c + sin b sin c cos A it becomes cos(n-A) = cos(n-B)cos(n-C)+sin(n-B)sin(n-C)cos(n-a) (A-14) or cos A =- cos B cos C + sin B sin C cos a which is one of the three Laws of Cosines for Angles. Similarly Equations (A-l0) and (A-ll) become sin A cos b = cos B sin C + sin B cos C cos a (A-15) sin B cos a = cos A sin C + sin A cos C cos b (A-16) which are two more of the twelve Five-Element Formulae. Equations relationships. (A-8) to (A-16) represent one-third of the The other two-thirds are obtained by simultaneous 181 cyclic permutation of a, b, c and A, Bt C. For example. from Equation (A-B) sin a sin B = sin b sin A we obtain sin b sin C = sin c sin B sin c sin A = sin a sin C The entire set of 27 relations obtained this way are stated below. A.2 Summary of Relationships A.2.1 Law of Sines sin a sin B = sin b sin A sin b sin C = sin c sin B sin c sin A = sin a sin C A.2.2 Law of Cosines (sides) cos a = cos b cos c + sin b sin c cos A cos b = cos c cos a + sin c sin a cos B cos c A.2.3 = cos a cos b + sin a sin b cos C Law of Cosines (angles) cos A = - cos B cos C + sin B sin C cos a cos B = - cos C cos A + sin C sin A cos b cos C = - cos A cos B + sin A sin B cos c A.2.4 Four-Element Formulae cos A cos c = sin c cot b - sin A cot B cos B cos a = sin a cot c - sin B cot C cos C cos b = sin b cot a - sin C cot A cos B cos c = sin c cot a - sin B cot A 182 A.2.5 sin C cot B cos C cos a = sin a cot b cos A cos b = sin b cot c - sin A cot C Five-Element Formulae sin a cos B = cos = cos A = cos b sin c - sin b cos c cos A sin b cos C c sin a - sin c cos a cos B sin c cos a sin b sin a cos b cos C sin b cos A = cos a sin c - sin a cos c cos B sin c cos B = cos b sin a - sin b cos a cos C sin a cos C = cos c sin b sin A cos b = cos sin c cos b cos A B sin C + sin B cos C cos a sin B cos c = cos C sin A + sin C cos A cos b sin C cos a = cos A sin B + sin A cos B cos c sin B cos a sin C cos b sin A cos c = cos = cos = cos A sin C + sin A cos C cos b B sin A + sin B cos A cos c C sin B + sin C cos B cos a 183 APPENDIX B CANADIAN TIME ZONES / I'J 110 I I I 120 100 I (Canada Year Book 1978-79, Dept. of Supply and Services, Ottawa) 184 APPENDIX C EXCERPT FROM THE FOURTH FUNDAMENTAL CATALOGUE (FK4) EQUINOX AND EPOCH 1950.0 AND 1975.0 No. US3 I ct 6.92 Ko rJ'43'" 5!-IH 9 44 28.7 10 + 333!:41 + 332.1>87 3.6'-1.8 Go + 164.799 + 164.796 5. 20 Go 9 -13 5 2.35-1 9 ~-I 3J.553 9 45 27.4H 9 46 59.02 • 3.89 Fo I"···1 + 19"2J34 I.co I Car uS5 Dr 1369 UMa . " U!\la 9 47 949 9 48 9 49 9 49 9 So IHH 42.$:6 58.396 6.00 At. 162 G. Vel HZ Ko 371 /l Leo 4·10 Ko 9 49 55·434 9 51 20.659 373 183 G. Hya 5.16 lIo 1$ G. Sex 7.03 Ko Grb 1586 UMa 5.96 Ko 9 52 30.60: 9 53 41.360 9 53 38.894 9 54 5H07 9 53 57.5 21 9 56 10.786 9 54 37.760 9 56 9-479 1256 1257 372 19 LMi 5·19 FS 375 '1' Vel 3·70 B5 377 11 Ant 5·=5 Fo 9 56 43-347 9 57 47.7 2 9 376 u Sex 6.63 A5 9 57 7·479 9 58 25.259 378 :r 4-89 Mo 1258 20 I.,.:i 5.60 GS 1259 Pi 9h229 UM,. 5·74 FS 1200 193 G. Hya 5·80 Fo 1261 ,,' Hy3 4·72 D8 ·379 11 Leo 3.5 8 Aop 380 Leo IE D8 1·34 :;81 1. lIya 3·S3 Ko :;85 (Q Car 3.56 118 4·09 A2 s Fo 382 19' G. Vel 3~4 C JeD 3 83 i. :262 \J!\b :;2 e I~G.l Ul\b. S:>t 187 G. C3. uli4 uGS )'J I G ••\ .. t 3· G I HZ A: 5·74 A3 Fo 5.40 KS 3·44 $.(,2 I D!/ + + + + 13.441 21·571 374 L~o + + + + 27-'02 6Su: 370 d'ct .~. .iii dT " J,. d7' 1-:1"(%) 1>1 (2) r>, (,.) I 125-1 368 dx sp. Name 9 55 6.~49 9 55 58.954 10 IZ 10 13 10 I:l 10 '3 10 '] 10 IS 10 14 10 I:: :6.08 :r.~ u 17.29 1·4 9 - 0.380 - 0.380 -0.001 -0.001 21 ..$3 7.6 40 - 0·969 - 0.963 + 0.168 + 0.174 - 1·599 - 1·591$ +0.004 +0.00-1 1-1·55 1·4 6 - 0.358 0·359 0.166 0.166 -0.003 -0.003 0.000 0.000 19.27 2.1 13 z6..p 2 .. ; 16 .- 1·733 1.72 5 1.056 1.055 +0.031 +0.032 20·57 4.6 21 +0.006 +0.006 20.46 :Z.o 10 - O.I]~ - 0.13$ - 0.806 - 0.808 -0.001 -0.001 13-38 5.6 SO -0.006 -0.006 18.47 3.6 ZJ +0.001 u·33 1·9 II 23 2.';00 232•643 210·9~1 + 14 2 .506 I -II -0.001 -0.001 3°2·309 302 •25 1 + 14 2 .7°3 10 15 .19·9'l-'I 10 !6 SS.69S 9·,; 10 S·t)9S - - 0.039 - 0.035 -10.691 -10.523 - 1.767 - J.7S$ + 0.487 + 0.495 + 0.438 + 0·445 - 0.276 - 0.271 - 0·393 - 0.389 ' - 1.230 - 1.223 - 3.066 - 3.044 + 0.:8$ + 0.29-1 + 0.086 + 0·09-' - 0.637 - 0.6]2 - 0.489 - 0 ...8.. + 0.080 ·t 0.<>S5 + + 0.39 1 0·:;97 + 333.6 .: + 333. 184 + 361 •689 + 360.7;'; - o.SSS 0.85 2 + ·435..B'; + H2.080 - 5.54 2 251.899 2';~.:Ot; 0.(109 0.618 1.877 I.S04 - 0.-163 - 0..;63 oj·O.OOI - 0. 21 5 .- 0.21 5 -0.001 -0.001 13.03 1.2 5 - '40.006 +0.006 24·37 2.5 II +0.003 +0.003 -0.002 -0.003 0.000 0.000 36.4 1 :;·9 19 19.07 2·9 IS 12.88 1.8 10 0.000 16.51 1·4 6 4·14.0 4·139 0. 26 9 0.26S - 0.845 - 0.846 - 0. 289 - 0.259 - o.OIZ - 0.0:2 - 1.6g6 +0.003 +0.00] 08.Sz 1.0 4 - 1.';07 - 1.40S -0.006 -0.006 15·45 I.,: 7 - 0.6:5 - 0;6:9 -0.014 -0.014 11.62 9·:; 49 - 1·345 -o.ooS 16·5; 4·-1 zo -0.001 -0.001 IS·9i 1·4 ,. 1.5 12 1·509 1_3S0 +0.010 -10.010 IS·9S 1·9 S +0.024 :13.4 2 4., IS - 1·3;4 -It'.?:" 17· I g I., 9 -,0.0'>1 -0.0.'16 -0.006 I·P9 9·! 39 0.0<» 19·1>~ s·.:! IS + + - 1·34; o.IH 0. 134 + 29S.118 ~. =9S.I:5 ~. 0."11j - 1.104 - 1.1::-'; + :00.367 + o·S~7 - 0·3.3 :oo.60~ oj. v.6oS + :;5.0 39 + .•. 27).2,)4 + 0·433 O~':.!5 0.000 1.695 - - 5·475 + 0.(111 + 10..l2 z.6 33.0-17 is IS 16,50 + 32 7.034 + 326.7 1; + 319.4 62 + 3 1 9.21 9 + 29 2.48, + 29 2 .5:: 1<' 24.581 10 16 14.il0 -0.003 -0.003 -0.0:6 -0.0:6 z.S +0.032 +0.033 + 397·034 + 277. 137 + 277.283 + 29 2• 143 + 292.IS7 ).353 3S·"SS 10 14 25.657 10 16 14.171 10 IS 8.659 10 If> 23.1)19 0·%34 0.235 l!.l6: 2.156 2j·H - 3·817 - 3.809 + 0.057 + 0.05i· + .H5·802 + 345. 11>9 + 398.561 of· + + -O!OOI 3·9S5 3-955 0.118 0.111 + 0.48: + 0.490 + :;16.823 + -0.001 426.403 42-1.418 + 317.018 37.998 4 1.011 54.75S Ig.107 + o!.60 + 0.160 - :.201 - 2.190 + 257·';'9 + 257.6]9 + 3 11 • 188 + 3".051 9 57 34·334 9 58 53.564 9 58 8.'.115 9593H89 10 I 17·747 10 2 57.196 10 2 2.261 10 3 11.563 10 :: 41.299 :0 :; 54.340 10 4 36.533 10 S 58•252 10 S 42.647 10 7 2·482 10 S 8.945 10 9 22.070 O!i09 0.70 5 0.0<J6 0.005 386.859 385.7 01 + 341.140 + ]40.657 + 282.989 + 283.075 + 298.06: + 298.043 . + 535.7 18 + 530.415 + 367.318 + 366.437 + zlo·696 + - - 0·3.4 - 0.116 - ".IIG -O.OO~ -n.<J?1 0.000 I 185 EQUINOX AND EPOCH 1950.0 AND 1975.0 li US3 u.ss 370 371 373 1257 372 ~j4 375 377 12 59 1260 +JSo 5';' +18 47 -62 16 -62 23 53~4l 57.86 3"..;% 3 1·9S +46 IS 18.ll +46 8 li.i2 +59 16 30.31 +59 9 25.83 - ( o 29.53 -4 7 32•11 -45 -46 +26 +26 -18 -18 35.21 36•08 3 1•2 i 18.u 25-34 31.32 44.86 . 53.70 -35 39 3. 66 -35 46 15.14 + 3 37 28.41 + 3 30 Ij·38 +8 17 5-7 6 +8 9 53. 13 +3 2 10 13.66 +3 2 2 50.19 +54 8 4·53 .+54 o 4S.15 -24 :I 3-\.25 -24 9 50·53 379 +17 o 26"7 + 16 53 6.45 +u 12 44.54 +12 5 23·90 -12 6 22.55 -12 13 48.08 .:62 +43 +43 +65 .. 65 ..- 7 -7 H =9·0~ -28 51 39·5·!· ,n (,,') GC N30 '4·0 26 Int-!- 2335 + + 0·33 0·33 +C.02 11.64 5·9 :15 13.162 233'> +0.02 9.53 9·S6 -0.18 -0.17 19.20 3. 2 12 13';97 :1345 - 15-55 - 15.48 3. 18 3.18 + 3.03 + 3.0'; +0.30 +0·:19 0.00 0.00 09·5' 2.2 8 13540 2355 17·74 2·4 13 13558 2357 +0.03 +0.03 +0.12 +0.12 20.6') 7.2 40 13574 2362 08.03 2.1 8 13590 :l36.; +0.03 +0.03 +0.01 "0.01 +0.13 "0.13 +0.08 ..0.08 +0.01 +0.01 21.81 3.8 21 136.&4 2367 24·43 4·1 26 13674 2371 14-8, 2·5 12 13684 2373 16-42 :l·7 lZ 13700 :1376 10.61 4.6 22 13711 237'1 +0.06 +0.06 19·37 4-5 :z6 13741 2387 19.14 3·3 16 13746 2389 09.71 1·9 7 13755 2390 17-73 3. 2 13 13763 :l391 26.'7 44 .6 1382 7 2397 + + -- 5·99 5.96 4·22 4·21 0.11 O.ll 4.05 4·0:1 :1·91 2·89 0.46 0·46 2.4 1 2·39 1.6') -11·39 -1l.26 -- :l.64 -12.25 -12·09 -14·00 -13.76 - 43.3 2 - 43. 24 0.66 0.66 +0.32 +0.32 +0.02 +0.02 -1742 .75 -1747·5° -1746.4= -175'.4 2 -1756.12 -176,.65 - 9·5-\ - 9.46 -.0.05 - 9·95 -11.12 -10·99 + + + + 1.83 1.85 +0.06 +0.06 +0.0% +0.02 0.00 0.00 18.20 4.0 :6 138.;8 '402 12·09 2·7 14 13861 z¥>8 13.83 :l.2 9 13899 :l41:1 -1759·9' -10·70 -10·57 + 01·35 ··7 5 13926 24'4 -1779·73 - 9·57 - 9·47 - 4.3 0 - 4. 27 - 7. 86 - i.SQ -10.48 -10·33 -11.32 -11.13 -13.65 -13-36 - 9. 18 - 9·07 +0.12 +0.12 +0.10 +0.10 14·0 j :l·3 II 13982 2424 +0.04 +0.0.0 12·97 4·6 23 14074 2434 +0·09 +0·09 -0.01 -0.01 14.27 4·7 22 14076 ::1435 '5·37 2.2 10 14107 ::144° 4.26 4. 2 4 1.19 1.17 010 +0.10 +0.10 06.9 1 2·4 S 14113 244% +0·09 13.14 ].0 12 14123 '445 16.67 2·7 IS '4129 2446 o.oS +0,"7 13.8., 5·9 :l4 14133 2448 4·5 25 14144 =45° -1787-70 -1789.84 -1784047 -li88.39 -1794·55 -1799·75 -179~29 -· ISo3·90 -1;96·54 -IS"3·30 -1798. 22 -1~":·iS '-17~~.~t; -11'::>1·:-7 I ("> +0.03 +0.03 +0.02 +0.02 -li8.;·49 52 25. 2 3 59 51•84 40 1-9~ 3 2 3~·65 953·H 2 23.:6 %1 31.58 14 I.h-;) 49 6.;8 56 36.91 711 26.00 - - 7·59 - 7·55 - 9. 21 - 9. 15 -11.20 -1I.oS - l i65· 2 3 -61 .. 5j·O'\ .. 61 I: 25. 1; -·as -19·98 -19·45 -13·5:1 -13·33 EI" (J) -0!01 -0.01 ----- -12·94 -1%·i9 -10.51 -10,42 -11.01 -10·90 'rii .- l!i6· ,.j6 -- - 8·i6 - 8.7 1 dl,I --- -11·59 -11·48 -1721·34 -17 2 6.91 -1727.67 -1733·33 -177°·84 -1776.92 -174 2 .04 -1748.gS -69 47 :!J·35 I" - 6.36 - 6·33 -15.36 -15. 14 -16..;8, -16.20 -1713.48 -1717. 2 6 -17 2 3.64 -1728.23 -69 54 4"·54 -41 -4' +23 +%3 -13!JI -13. 17 -17°7.41 -17 12•89 -1712.77 -17 22•63 -1714.6') -1721.40 40.83 -12 49 'j·50 -12 5634-73 - 16SI!!9 1 -1665·53 -1650.65 -1663.82 -1677.S:, - 1685.44 -1696.01 -17°2·44 -1706.39 -17".5 2 P7 5i·74 1261 ! iii>! -1687·48 -1 693-25 -1684·48 -1688.s,. - 7 24 26·90 -7 31 34·44 "73 7 +7:1 59 "41 17 +4 1 10 -S.J 19 -54 26 dT' -1693.85 -17°2.02 57 33·54 4 14 7 46 53 d:') ,111 ~1;9"·92 ":&-'1. 1 1 - ... 6.00 ! - S·';3 - S.41 1·70 2.6.J -- -+ - 0·95 0.96 0.58 0.58 0.26 0.29 9.46 -:- 9·44 + + + + .----+ + 6.04 • I + + I I 0.29 0·3° 3·84 3. 86 1.21 1.21 +0·09 +o.Oj o·~9 +o.o~ 0.30 01- 0.8S + 0.5S ..... O.Ol ·to.OI ;C.OI 19."') I I I I 186 APPENDIX D EXCERPT FROM APPARENT PLACES OF FUNDAMENTAL STARS APPARENT PLACES OF STARS, 1981 AT UPPER TRANSIT AT GREENWICH ---·--T---------------r--------------.--------------~--------------- 379 1261 No. Name IX Tlleonis " Hydrae 381 380 Leonis i. Hydraa (Regulus) Mafl·SpeeL 4.7.2 SS 3.58 AOp 1.34 88 3.83 KO U.T. RA Dec. R.A. Dec. R.A. Dec. RA Dec. b m 1004 d 1 -7.8 1 2.1 1 12.1 1 22.1 2 1.1 :2. 11.0 2 21.0 3 3.0 3 12.9 3 22.9 + JOO S 11.861 .~ 12.145 .. 252 12.397 + 2C9 12.600 .. 166 12.772 12.888 12.955 12.976 12.952 12.001 h 0' -1258 -23) S .. + 16 50 3Zl 14.55 -m 17.535 .. 304 76.96 -157 -136 16.94 -240 17.839 • 273 75.60 -112 19.34 -233 18.112 + 232 74.48 _ 84 21.67 -220 1S".344 .. 187 73.64 - 56 23.87 18.531 73.08 -203 .. 116 .. 67 25.90 -100 18.~7 • 21 27.70 -157 18. i51 _ 24 CJ.27 -130 18.786 18.774 _ 61 30.57 31.59 -102 18.722 ,",,, -n .. 136 .. 84 .. 35 72.82 - 26 .. 2 72.84 _ 2~ 73.08 • .:s - 12 73.54 .. &l - 52 74.14 - 64 5 1.8 5 11.8 12.802 _ 115 "".36 • 49 18.638 _ 111 74.85 : ~ 12.687 _ 128 32.85 _ 24 18.527 _ 126 75.63 + 79 12.559 137 33.09 _ 1 18.401 _ 134 76.42 ·78 12.422 33.10 18.267 _ 139 77.20 .. 74 12.282 - 140 32.86 .. 24 18.128 77.94 5 21.8 5 31.7 G 10.7 I; 20.7 6 30.6 '2.148 12.020 11.905 11.806 11.723 1 10.6 7 20.6 1 30.6 11.660 _ 11 621 _ 4· 1.9 4 11.9 4 21.8 8 9.5 8 19.5 8 29.5 9 8.5 9 18.4 9 28.4 10 8.4 - 89 - 134 - 128 _ 115 _ 99 _ 83 - 63 39 17 11.604.. 9 11.613 .. 38 11.651 11.717 11.816 11.947 12.112 12.313 32.40 .. 66 17.997 =:~ 31.74 -as 17.874 _ 110 30.88 -102 17.764 29.86 +116 17.674 - 90 17.601 - 73 28.70 +46 +128 ... .. +115 .. .. 89 20.94 .. 9S 17.673 .. 131 19.99 .. 10 17.784 .. 165 19.29 .. 42 17.925 .. :<Dl 18.87 .. 7 18.098 18.306 18.80 .. Zl2 - 27.42 +133 17.551 26.09 .138 17.525 24.71 +135 17.521 23.36 -127 17.545 17.595 22.00 .. 66 h mOl 10 06 .. .. • .. 78.60 79.18 79.66 80.03 80.30 ., .. .. ... m 10 01 -171 2'"350 • . 21.647 21.914 22.140 22.322 22.454 22.535 22.570 22.557 22.507 22.425 22.317 22.194 22.063 21.928 66 58 48 37 21.800 .21.680 21.573 • 27 21.483 21.411 50 .. 14 21.361 26 80.44 + 0 4 80.44 .• 12 21.334 24 80.32 - 28 21.328 50 80.04 - 44 21.349 79.60 21.397 78 9 - a:: 111 7.00 _ 79 21.472 141 78.21 _ 93 21.578 173 77.23 -116 21.714 2C8 76.07 -135 21.883 74.72 22.085 317 • 297 m h 0' + 12 03 10 09 S - :m ", -12 15 35.65 -155 39.573 237 29.85 .267 34.10 -135 39.800 : eli 32.21 ~ 226 32.75 -110 40.115 • 215 34.59 • 182 31.65 _ 64 40.330 • 170 36.89 39.07 30.81 4O.ECO .. 1!2 -m '" 40.622 • lZ2 ... 81 30.26 ~ 72 41.07 -m .. 35 29.97 - 5 40.694 • 27 42.84 -Is:! - 13 29.92 • 18 40.721 _ 19 44.37 -128 - 50 30.10 • 3' 40.702 _ ::0 45.65 -IOJ 40.647 30.44 46.65 -- - 82 _ 100 30.92 _ 123 31.51 _ 131 32.16 135 32.83 33.51 ·48 • 59 _ €C • 67 • €a .• ;28 - 120 - 107 - 90 - 72 34.16 ..• as 62 34.78 .. 55 35.34 • EC 35.84 .. t' , 36.27 - 50 -3.; 36.61 • Zl - 6 36.84 .. 13 .. 21 36.97 _ 1 .. 48 36.96 - ~6 36.80 - 27 i'5 -!2 ,;::.0 35.97 _ 71 + • '00 36.48 - 51 oj. ·-84 - 110 - l:iS - 134 - 138 40.563 40.453 40.328 40.194 40.0"...0 39.923 39.796 39.680 39.E80 39.496 39.431 47.40 -49 47.88 -23 48.11 • a 48.11 • 23 47.88 - 133 _ 127 47.43 -63 - 116 46.80 • 8-1 _ 100 45.96 • sa _ o. 44.98 .m "'" 43.87 22 42.63 41.35 6 32 38.73 37.52 =: 39.388 3936639372: 39.404 39.465 • 39 1':59 • 4O.m 61 !i4 .. 126 '110 36.42 -00 35.52 34.80 34.43 34.44 3188 ·t:;:: '".2''''' +. ~ 34 6 . -153 '+V uo ""'" .7 30.35 -tea 40.528 • «a 35.45 .", _ 163 35.26 _ 92 39.685 • 1~ _ 202 34.34 ,,- 39.844 11:'6 33.21 - • 40 Q4{)'. 10 10 11 11 11 18.3 28.3 7.3 17.3 272 12.545 + 264 12.809 .. 230 13.COO .. :lOO 13.408 .. 323 13.731 • 239 7321 -151 22319 • 23-1 19.10 - 67 18545 . • 211 . -166 . • 264 19.77 -107 18.816 .298 71.55 -m 22.583 • 293 20.84 -142 19.114 • 319 69.78 -182 22.876 .311 28.67 -tIS 40.816 • X6 36.53 22.26 -175 19.433 • 334 67.96 -las 23.187 • 328 26.89 -is:: 41.122 • 323 37.96 24.01 19.767 66.11 23.515 25.03 41.445 39.70 12 12 12 12 7.2 17.2 27.2 37.1 14.057:: 14.375 .. 303 14.678 .. 274 14.952 .. 239 26.03' 28.25 -230 30.61 -241 33.02 -238 20.108 20.443 20.765 21061 ______ ____ ~,. --~~ =: __ 11.872 -'.026 19.85 -0.230 17.852 OO('I'~ dli(1;1) +() 0"'...3 date). di{<l -0.013 '-0.35 -0,48 -<l.C65 -0018 february 21 64.32 62.65 61.13 59.85 ow • • 33 ~ - 32 S -H:6 -1£3 -1;! ::~ 23.B4S : : 23.18 ::: 41.771 :;: 41.72 =~ 24.177 + 315 21.39 -16; 42.000 • 1:6 43.92 -Z3-l 24.492 • 2S9 19.72 -14S 42.395 • 278 46.26 -2!O =,02 24.781 • 255 18.23 -127 42.673 • :14: 48 66 -:35 1:.2 ',28 --~~----~~----~+_----~~----~~----~L-----~ Me." 1'1""" see 0, tanS Db1e.Tran•. : ~ • 322 • 296 • 261 ::~ ·1.045 80.71 -<l.303 21.633 -1023 37.95 -<l214 39 622 -1023 35 i8 -0.35 -<l.004 -<l013 -0.35 -o.OSS '0.47 -0013 -0.35 -<l46 -<l.4B February 21 February 22 -0.217 February 22